{"id":29365,"date":"2023-05-12T05:34:17","date_gmt":"2023-05-12T05:34:17","guid":{"rendered":"https:\/\/www.splashlearn.com\/math-vocabulary\/?page_id=29365"},"modified":"2024-01-16T10:15:40","modified_gmt":"2024-01-16T10:15:40","slug":"45-45-90-triangle-properties-formulas-construction","status":"publish","type":"post","link":"https:\/\/www.splashlearn.com\/math-vocabulary\/45-45-90-triangle","title":{"rendered":"45\u00b0-45\u00b0-90\u00b0 Triangle: Properties, Formulas, Construction"},"content":{"rendered":"<div class=\"aioseo-breadcrumbs\"><span class=\"aioseo-breadcrumb\">\n\tMath-Vocabulary\n<\/span><\/div>\n\n<div class=\"ub_table-of-contents\" data-showtext=\"show\" data-hidetext=\"hide\" data-scrolltype=\"auto\" id=\"ub_table-of-contents-de5bf717-6aaa-43e3-a724-81832a2dc1ea\" data-initiallyhideonmobile=\"false\"\n                    data-initiallyshow=\"true\"><div class=\"ub_table-of-contents-extra-container\"><div class=\"ub_table-of-contents-container ub_table-of-contents-1-column \"><ul><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/45-45-90-triangle#0-what-is-a-45%C2%B0-45%C2%B0-90%C2%B0-triangle>What Is a 45\u00b0-45\u00b0-90\u00b0 Triangle?<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/45-45-90-triangle#3-how-to-solve-a-45%C2%B0-45%C2%B0-90%C2%B0-triangle>How to Solve a 45\u00b0-45\u00b0-90\u00b0 Triangle<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/45-45-90-triangle#4-45%C2%B0-45%C2%B0-90%C2%B0-triangle-formulas>45\u00b0-45\u00b0-90\u00b0 Triangle Formulas<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/45-45-90-triangle#9-solved-examples-on-the-45%C2%B0-45%C2%B0-90%C2%B0-triangle>Solved Examples on the 45\u00b0-45\u00b0-90\u00b0 Triangle<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/45-45-90-triangle#10-practice-problems-on-45%C2%B0-45%C2%B0-90%C2%B0-triangle>Practice Problems on 45\u00b0-45\u00b0-90\u00b0 Triangle<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/45-45-90-triangle#11-frequently-asked-questions-on-45%C2%B0-45%C2%B0-90%C2%B0-triangle>Frequently Asked Questions on 45\u00b0-45\u00b0-90\u00b0 Triangle<\/a><\/li><\/ul><\/div><\/div><\/div>\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"0-what-is-a-45%C2%B0-45%C2%B0-90%C2%B0-triangle\">What Is a 45\u00b0-45\u00b0-90\u00b0 Triangle?<\/h2>\n\n\n\n<p class=\"eplus-wrapper\"><strong>The <\/strong>$45^\\circ\\;-\\;45^\\circ\\;-\\;90^\\circ$<strong> triangle, as the name itself suggests, is a triangle in which there is one <\/strong>$90^\\circ$<strong> angle (right angle), and two <\/strong>$45^\\circ$<strong> angles.&nbsp;<\/strong><\/p>\n\n\n\n<p class=\"eplus-wrapper\">Since two angles are congruent, the sides opposite to the congruent angles are also congruent. Two congruent sides and one right angle make the $45^\\circ\\;-\\;45^\\circ\\;-\\;90^\\circ$ triangle an isosceles right triangle.&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper\">These triangles have special significance in geometry because they follow properties of right triangles as well as isosceles triangles.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full eplus-wrapper\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"660\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/45\u00b0-45\u00b0-90\u00b0-triangle.png\" alt=\"45\u00b0-45\u00b0-90\u00b0 triangle\" class=\"wp-image-35712\" title=\"45\u00b0-45\u00b0-90\u00b0 triangle\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/45\u00b0-45\u00b0-90\u00b0-triangle.png 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/45\u00b0-45\u00b0-90\u00b0-triangle-282x300.png 282w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n\n\n\n<p class=\"eplus-wrapper\">$45^\\circ\\;-\\;45^\\circ\\;-\\;90^\\circ$<strong> Triangle: Definition<\/strong><\/p>\n\n\n\n<p class=\"eplus-wrapper\"><strong>A <\/strong>$45^\\circ\\;-\\;45^\\circ\\;-\\;90^\\circ$<strong> triangle is an isosceles right triangle<\/strong>. It is a special type of right triangle in which the three interior angles are $45^\\circ,\\; 45^\\circ,$ and $90^\\circ$.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full eplus-wrapper\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"619\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/45\u00b0-45\u00b0-90\u00b0-triangle-ABC-its-legs-and-hypotenuse.png\" alt=\"45\u00b0-45\u00b0-90\u00b0 triangle ABC, its legs and hypotenuse\" class=\"wp-image-35713\" title=\"45\u00b0-45\u00b0-90\u00b0 triangle ABC, its legs and hypotenuse\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/45\u00b0-45\u00b0-90\u00b0-triangle-ABC-its-legs-and-hypotenuse.png 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/45\u00b0-45\u00b0-90\u00b0-triangle-ABC-its-legs-and-hypotenuse-300x300.png 300w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/45\u00b0-45\u00b0-90\u00b0-triangle-ABC-its-legs-and-hypotenuse-150x150.png 150w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/45\u00b0-45\u00b0-90\u00b0-triangle-ABC-its-legs-and-hypotenuse-250x250.png 250w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/45\u00b0-45\u00b0-90\u00b0-triangle-ABC-its-legs-and-hypotenuse-120x120.png 120w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n\n\n\n<p class=\"eplus-wrapper\">The side opposite to the right angle is called hypotenuse. It is the longest side of any right triangle. Here, AC is the hypotenuse. It is the longest side of the triangle.&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper\">The sides opposite to the $45^\\circ$ angles are congruent. They are also known as legs of the right triangle. Here, $AB = BC$.<\/p>\n\n\n\n<div id=\"recommended-games-container-id\" class=\"recommended-games-container\"><h4 class=\"recommended-games-container-headline\">Recommended Games<\/h4><div class=\"recommended-games-container-slides\"><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/answer-questions-related-to-triangles\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/geometry_classify_tri_on_side_pt.png\" alt=\"Answer Questions Related to Triangles Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Answer Questions Related to Triangles Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/classify-triangles-and-rectangles-as-closed-shape\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/geometry_triangle_rectangle_2_pt.png\" alt=\"Classify Triangles and Rectangles as Closed Shape Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Classify Triangles and Rectangles as Closed Shape Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/classify-triangles\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/geometry_classify_triangles_pt.png\" alt=\"Classify Triangles Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Classify Triangles Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/identify-the-type-of-triangles-based-on-angles\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/geometry_classify_tri_on_angle_pt.png\" alt=\"Identify the Type of Triangles Based on Angles Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Identify the Type of Triangles Based on Angles Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/identify-triangles-and-rectangles\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/geometry_amusement_park_18_20_gm.png\" alt=\"Identify Triangles and Rectangles Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Identify Triangles and Rectangles Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/identify-triangles-and-squares\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/geometry_triangles_squares_pt.png\" alt=\"Identify Triangles and Squares Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Identify Triangles and Squares Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/identify-types-of-triangles\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/geometry_types_of_triangles_pt.png\" alt=\"Identify Types of Triangles Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Identify Types of Triangles Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/match-triangles-and-squares\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/geometry_amusement_park_4_5_gm.png\" alt=\"Match Triangles and Squares Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Match Triangles and Squares Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/triangles-and-rectangles\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/geometry_triangle_rectangle_1_pt.png\" alt=\"Triangles and Rectangles Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Triangles and Rectangles Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><\/div><p class=\"recommended-games-container-desc\"><a href=\"https:\/\/www.splashlearn.com\/games\">More Games<\/a><\/p><button class=\"scroll-right-arrow\"><\/button><\/div><script type=\"text\/javascript\">\n        document.addEventListener(\"DOMContentLoaded\", function() {\n            const container = document.querySelector(\"#recommended-games-container-id\");\n            const slidesContainer = container.querySelector(\".recommended-games-container-slides\");\n            const cards = slidesContainer.querySelectorAll(\".game-card-container-outer\");\n            const scrollRightArrow = container.querySelector(\".scroll-right-arrow\");\n\n            function adjustContainerStyles() {\n                const numCards = cards.length;\n\n                if (numCards === 1) {\n                    container.style.maxWidth = \"30%\";\n                    container.style.textAlign = \"center\";\n                } else if (numCards === 2) {\n                    container.style.maxWidth = \"50%\";\n                    container.style.textAlign = \"center\";\n                } else if (numCards === 3) {\n                    container.style.maxWidth = \"75%\";\n                    container.style.textAlign = \"center\";\n                } else {\n                    container.style.maxWidth = \"\";\n                    container.style.textAlign = \"\";\n                }\n            }\n\n            function checkScrollPosition() {\n                const maxScrollLeft = slidesContainer.scrollWidth - slidesContainer.clientWidth;\n                if ((slidesContainer.scrollLeft + 10) >= maxScrollLeft) {\n                    scrollRightArrow.style.display = \"none\"; \/\/ Hide the arrow if fully scrolled\n                } else {\n                    scrollRightArrow.style.display = \"block\"; \/\/ Show the arrow if not fully scrolled\n                }\n            }\n\n            scrollRightArrow.addEventListener(\"click\", function() {\n                const scrollAmount = 300; \/\/ Adjust based on the container's width and your needs\n                slidesContainer.scrollLeft += scrollAmount;\n                setTimeout(checkScrollPosition, 100); \/\/ Delay to allow scroll update\n            });\n\n            adjustContainerStyles();\n            checkScrollPosition();\n            slidesContainer.addEventListener(\"scroll\", checkScrollPosition);\n        });\n    <\/script><h2 class=\"wp-block-heading eplus-wrapper\" id=\"1-45%C2%B0-45%C2%B0-90%C2%B0-triangle-theorem\">45\u00b0-45\u00b0-90\u00b0 Triangle Theorem<\/h2>\n\n\n\n<p class=\"eplus-wrapper\"><strong>The 45\u00b0-45\u00b0-90\u00b0 triangle theorem states that the length of the hypotenuse of a <\/strong>$45^\\circ\\;-\\;45^\\circ\\;-\\;90^\\circ$<strong> triangle is equal to square root times the length of a leg.<\/strong><\/p>\n\n\n\n<p class=\"eplus-wrapper\">Consider a<strong> <\/strong>$45^\\circ\\;-\\;45^\\circ\\;-\\;90^\\circ$ triangle ABC.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full eplus-wrapper\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"351\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/side-lengths-in-45\u00b0-45\u00b0-90\u00b0-triangle.png\" alt=\"Side lengths in 45\u00b0-45\u00b0-90\u00b0 triangle\" class=\"wp-image-35715\" title=\"Side lengths in 45\u00b0-45\u00b0-90\u00b0 triangle\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/side-lengths-in-45\u00b0-45\u00b0-90\u00b0-triangle.png 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/side-lengths-in-45\u00b0-45\u00b0-90\u00b0-triangle-300x170.png 300w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n\n\n\n<p class=\"eplus-wrapper\">Suppose that the lengths of the congruent sides (legs) triangle is x.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Thus, we have&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$AB = x$ and $BC = x$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">We can find the length of the hypotenuse AC using Pythagoras\u2019 theorem.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$AC^2 = AB^2 + BC^2$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$AC^2\\; Hypotenuse^2 = x^2 + x^2 = 2x^2$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Thus, length of the hypotenuse $= AC = \\sqrt{2x}$<\/p>\n\n\n\n<div id=\"recommended-worksheets-container-id\" class=\"recommended-games-container\"><h4 class=\"recommended-games-container-headline\">Recommended Worksheets<\/h4><div class=\"recommended-games-container-slides\"><div class=\"worksheet-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-worksheets\/division-by-9-within-90\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"worksheets_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t    <div class=\"worksheet-card-container-inner-block\">\r\n\t        <img decoding=\"async\" class=\"worksheet-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/cms_assets\/s\/math-worksheets\/division-by-9-within-90.jpeg\" alt=\"Division by 9 within 90\">\r\n\t    <\/div>\r\n\t\t<div class=\"worksheet-card-container-inner\" >\r\n\t\t<\/div><\/a>\r\n<\/div><div class=\"worksheet-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-worksheets\/factors-of-45\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"worksheets_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t    <div class=\"worksheet-card-container-inner-block\">\r\n\t        <img decoding=\"async\" class=\"worksheet-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/cms_assets\/s\/math-worksheets\/factors-of-45.jpeg\" alt=\"Factors of 45\">\r\n\t    <\/div>\r\n\t\t<div class=\"worksheet-card-container-inner\" >\r\n\t\t<\/div><\/a>\r\n<\/div><div class=\"worksheet-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-worksheets\/rectangles-and-triangles\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"worksheets_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t    <div class=\"worksheet-card-container-inner-block\">\r\n\t        <img decoding=\"async\" class=\"worksheet-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/cms_assets\/s\/math-worksheets\/rectangles-and-triangles.jpeg\" alt=\"Rectangles and Triangles Worksheet\">\r\n\t    <\/div>\r\n\t\t<div class=\"worksheet-card-container-inner\" >\r\n\t\t<\/div><\/a>\r\n<\/div><div class=\"worksheet-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-worksheets\/separate-out-the-triangles\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"worksheets_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t    <div class=\"worksheet-card-container-inner-block\">\r\n\t        <img decoding=\"async\" class=\"worksheet-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/cms_assets\/s\/math-worksheets\/separate-out-the-triangles.jpeg\" alt=\"Separate Out the Triangles Worksheet\">\r\n\t    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         }\n            }\n\n            scrollRightArrow.addEventListener(\"click\", function() {\n                const scrollAmount = 300; \/\/ Adjust based on the container's width and your needs\n                slidesContainer.scrollLeft += scrollAmount;\n                setTimeout(checkScrollPosition, 100); \/\/ Delay to allow scroll update\n            });\n\n            adjustContainerStyles();\n            checkScrollPosition();\n            slidesContainer.addEventListener(\"scroll\", checkScrollPosition);\n        });\n    <\/script><h2 class=\"wp-block-heading eplus-wrapper\" id=\"2-45%C2%B0-45%C2%B0-90%C2%B0-triangle-ratios\">45\u00b0-45\u00b0-90\u00b0 Triangle Ratios<\/h2>\n\n\n\n<p class=\"eplus-wrapper\">The ratio of side lengths in a $45^\\circ\\;-\\;45^\\circ\\;-\\;90^\\circ$ triangle $= AB : BC : AC$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">&nbsp;$= x : x : \\sqrt{2x}$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">&nbsp;$= 1 : 1 : \\sqrt{2}$<\/p>\n\n\n\n<p class=\"eplus-wrapper\"><strong>Ratio of angles in a <\/strong>$45^\\circ\\;-\\;45^\\circ\\;-\\;90^\\circ$<strong> triangle is <\/strong>$45^\\circ: 45^\\circ: 90^\\circ = 1 : 1 : 2$<\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"3-how-to-solve-a-45%C2%B0-45%C2%B0-90%C2%B0-triangle\">How to Solve a 45\u00b0-45\u00b0-90\u00b0 Triangle<\/h2>\n\n\n\n<p class=\"eplus-wrapper\">To solve a $45^\\circ\\;-\\;45^\\circ\\;-\\;90^\\circ$ triangle means to find the length of the missing sides when the length of one side of the right triangle is given. We can find the triangle\u2019s perimeter and the triangle\u2019s area, too.&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper\">There are two cases that we consider.<\/p>\n\n\n\n<ol class=\"eplus-wrapper wp-block-list\">\n<li class=\"eplus-wrapper\">To calculate the hypotenuse when the length of one of the sides is given.<\/li>\n<\/ol>\n\n\n\n<p class=\"eplus-wrapper\">This can be calculated by multiplying the given side by 2.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Hypotenuse $= \\sqrt{2} \\times Leg = \\sqrt{2x}$<\/p>\n\n\n\n<ol start=\"2\" class=\"eplus-wrapper wp-block-list\">\n<li class=\"eplus-wrapper\">To calculate the length of the equal sides of the triangle when the length of the hypotenuse is given.<\/li>\n<\/ol>\n\n\n\n<p class=\"eplus-wrapper\">We know that the ratio of side lengths in a $45^\\circ\\;-\\;45^\\circ\\;-\\;90^\\circ$ triangle is $1 : 1 : \\sqrt{2}$. The length of the equal sides can be calculated by just dividing the length of the hypotenuse by 2.&nbsp;<\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"4-45%C2%B0-45%C2%B0-90%C2%B0-triangle-formulas\">45\u00b0-45\u00b0-90\u00b0 Triangle Formulas<\/h2>\n\n\n\n<p class=\"eplus-wrapper\">Suppose that the length of one leg in a $45^\\circ\\;-\\;45^\\circ\\;-\\;90^\\circ$ triangle is x. Let\u2019s discuss different formulas associated with this triangle.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full eplus-wrapper\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"458\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/45\u00b0-45\u00b0-90\u00b0-triangle-PQR-with-legs-x-and-hypotenuse-2x.png\" alt=\"45\u00b0-45\u00b0-90\u00b0 triangle PQR with legs x and hypotenuse 2x\" class=\"wp-image-35716\" title=\"45\u00b0-45\u00b0-90\u00b0 triangle PQR with legs x and hypotenuse 2x\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/45\u00b0-45\u00b0-90\u00b0-triangle-PQR-with-legs-x-and-hypotenuse-2x.png 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/45\u00b0-45\u00b0-90\u00b0-triangle-PQR-with-legs-x-and-hypotenuse-2x-300x222.png 300w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n\n\n\n<p class=\"eplus-wrapper\"><strong>Other Leg<\/strong><\/p>\n\n\n\n<p class=\"eplus-wrapper\">If the length of one leg is x units, then the length of the other leg will also be x units. Two legs of a $45^\\circ\\;-\\;45^\\circ\\;-\\;90^\\circ$ triangle are equal.<\/p>\n\n\n\n<p class=\"eplus-wrapper\"><strong>Hypotenuse<\/strong><\/p>\n\n\n\n<p class=\"eplus-wrapper\">The length of the hypotenuse a $45^\\circ\\;-\\;45^\\circ\\;-\\;90^\\circ$ triangle $= \\sqrt{2}\\times Side = \\sqrt{2x}$&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper\"><strong>Area<\/strong><\/p>\n\n\n\n<p class=\"eplus-wrapper\">We know that the area of triangle is calculated by the formula $\\frac{1}{2} \\times$Base $\\times$Height.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">In a <em>&nbsp;<\/em>$45^\\circ\\;-\\;45^\\circ\\;-\\;90^\\circ$ triangle, base $= x$, height $= x$.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Area of a $45^\\circ\\;-\\;45^\\circ\\;-\\;90^\\circ$ triangle $= \\frac{1}{2} \\times x \\times x = \\frac{x^2}{2}$<\/p>\n\n\n\n<p class=\"eplus-wrapper\"><strong>Perimeter<\/strong><\/p>\n\n\n\n<p class=\"eplus-wrapper\">The perimeter is the sum of the sides of the triangle.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Perimeter of a $45^\\circ\\;-\\;45^\\circ\\;-\\;90^\\circ$ triangle $= x + x + \\sqrt{2x} = x(2 + \\sqrt{2}$.<\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"5-45%C2%B0-45%C2%B0-90%C2%B0-triangle-rules-and-properties\">45\u00b0-45\u00b0-90\u00b0 Triangle Rules and Properties<\/h2>\n\n\n\n<ul class=\"eplus-wrapper wp-block-list\">\n<li class=\"eplus-wrapper\">It has one $90^\\circ$ angle and two $45^\\circ$ angles.&nbsp;<\/li>\n<\/ul>\n\n\n\n<ul class=\"eplus-wrapper wp-block-list\">\n<li class=\"eplus-wrapper\">The $45^\\circ\\;-\\;45^\\circ\\;-\\;90^\\circ$ right triangle is the only possible right triangle that is also an isosceles triangle.<\/li>\n<\/ul>\n\n\n\n<ul class=\"eplus-wrapper wp-block-list\">\n<li class=\"eplus-wrapper\">Sides opposite to the $45^\\circ$ angles are equal.<\/li>\n<\/ul>\n\n\n\n<ul class=\"eplus-wrapper wp-block-list\">\n<li class=\"eplus-wrapper\">The side opposite to the $90^\\circ$ angle is the hypotenuse and it is the longest side.<\/li>\n<\/ul>\n\n\n\n<ul class=\"eplus-wrapper wp-block-list\">\n<li class=\"eplus-wrapper\">Since there are two equal sides, the $45^\\circ\\;-\\;45^\\circ\\;-\\;90^\\circ$ triangle is an isosceles right triangle.<\/li>\n<\/ul>\n\n\n\n<ul class=\"eplus-wrapper wp-block-list\">\n<li class=\"eplus-wrapper\">The ratio of $45^\\circ\\;-\\;45^\\circ\\;-\\;90^\\circ$ triangle sides equals 1 : 1 : $\\sqrt{2}$.<\/li>\n<\/ul>\n\n\n\n<p class=\"eplus-wrapper\">The ratio of $45^\\circ\\;-\\;45^\\circ\\;-\\;90^\\circ$ triangle angles equals 1 : 1 : 2.<\/p>\n\n\n\n<ul class=\"eplus-wrapper wp-block-list\">\n<li class=\"eplus-wrapper\">This triangle has only one line of symmetry .<\/li>\n<\/ul>\n\n\n\n<figure class=\"wp-block-image size-full eplus-wrapper\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"473\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/one-line-of-symmetry-in-the-45\u00b0-45\u00b0-90\u00b0-triangle.png\" alt=\"One line of symmetry in the 45\u00b0-45\u00b0-90\u00b0 triangle\" class=\"wp-image-35717\" title=\"One line of symmetry in the 45\u00b0-45\u00b0-90\u00b0 triangle\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/one-line-of-symmetry-in-the-45\u00b0-45\u00b0-90\u00b0-triangle.png 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/one-line-of-symmetry-in-the-45\u00b0-45\u00b0-90\u00b0-triangle-300x229.png 300w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n\n\n\n<p class=\"eplus-wrapper\">We can see the line that passes through the vertex which has a 90-degree angle and is perpendicular to the opposite side is the line of symmetry because it acts as a mirror, i.e., the figure on either side of the line is identical.<\/p>\n\n\n\n<ul class=\"eplus-wrapper wp-block-list\">\n<li class=\"eplus-wrapper\">The $45^\\circ\\;-\\;45^\\circ\\;-\\;90^\\circ$ triangle does not possess a line of rotational symmetry.<\/li>\n<\/ul>\n\n\n\n<ul class=\"eplus-wrapper wp-block-list\">\n<li class=\"eplus-wrapper\">It has the smallest hypotenuse to the sum of the legs ratio.&nbsp;<\/li>\n<\/ul>\n\n\n\n<p class=\"eplus-wrapper\">It has the greatest altitude ratio from the hypotenuse to the sum of the legs.<\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"6-how-to-construct-a-45%C2%B0-45%C2%B0-90%C2%B0-triangle\">How to Construct a 45\u00b0-45\u00b0-90\u00b0 Triangle<\/h2>\n\n\n\n<p class=\"eplus-wrapper\">The $45^\\circ\\;-\\;45^\\circ\\;-\\;90^\\circ$ triangle with leg x is half of a square having side x. A square has four right angles. When it is cut diagonally, two right-angled triangles. Both angles are $45^\\circ\\;-\\;45^\\circ\\;-\\;90^\\circ$ triangles: one original $90^\\circ$ angle of the square, and the other two $90^\\circ$ angles bisected that give a $45^\\circ$ angle each.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">The diagonal of a square becomes the hypotenuse of a right triangle.<\/p>\n\n\n\n<p class=\"eplus-wrapper\"><strong>Step 1: <\/strong>Construct a square that has the same length as the length of the legs of the triangle we want to construct.<\/p>\n\n\n\n<p class=\"eplus-wrapper\"><strong>Step 2:<\/strong> Draw one diagonal of the square.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">This creates two congruent $45^\\circ\\;-\\;45^\\circ\\;-\\;90^\\circ$ triangles and the diagonal becomes the hypotenuse of the triangle.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full eplus-wrapper\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"516\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/45\u00b0-45\u00b0-90\u00b0-Triangle-is-half-of-a-square.png\" alt=\"45\u00b0-45\u00b0-90\u00b0 Triangle is half of a square\" class=\"wp-image-35718\" title=\"45\u00b0-45\u00b0-90\u00b0 Triangle is half of a square\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/45\u00b0-45\u00b0-90\u00b0-Triangle-is-half-of-a-square.png 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/45\u00b0-45\u00b0-90\u00b0-Triangle-is-half-of-a-square-300x250.png 300w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"7-facts-about-45%C2%B0-45%C2%B0-90%C2%B0-triangle\">Facts about 45\u00b0-45\u00b0-90\u00b0 Triangle<\/h2>\n\n\n\n<ul class=\"eplus-wrapper wp-block-list\">\n<li class=\"eplus-wrapper\">Carpenters, architects, and surveyors utilize right triangles to assure &#8220;square corners.&#8221;<\/li>\n\n\n\n<li class=\"eplus-wrapper\">All the $45^\\circ\\;-\\;45^\\circ\\;-\\;90^\\circ$ triangles are similar triangles. This is verified by the AA method of similarity which says that if two triangles have 2 angles of the same degree each they are similar.<\/li>\n\n\n\n<li class=\"eplus-wrapper\">This triangle can be viewed as exactly half of a square having each side length as the length of the legs of the triangle.<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"8-conclusion\">Conclusion<\/h2>\n\n\n\n<p class=\"eplus-wrapper\">In this article, we learned about the $45^\\circ\\;-\\;45^\\circ\\;-\\;90^\\circ$ triangle, its properties, construction, the ratio of interior angles, the ratio of sides, and different formulas associated with the triangle. Let\u2019s apply these concepts to solve a few examples and practice problems.<\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"9-solved-examples-on-the-45%C2%B0-45%C2%B0-90%C2%B0-triangle\">Solved Examples on the 45\u00b0-45\u00b0-90\u00b0 Triangle<\/h2>\n\n\n\n<ol class=\"eplus-wrapper wp-block-list\">\n<li class=\"eplus-wrapper\"><strong>&nbsp;The hypotenuse of a <\/strong>$45^\\circ\\;-\\;45^\\circ\\;-\\;90^\\circ$<strong> triangle is <\/strong>$4\\sqrt{2}$<strong> inches. Find the length of the base and height of the triangle.<\/strong><\/li>\n<\/ol>\n\n\n\n<p class=\"eplus-wrapper\"><strong>Solution:&nbsp;<\/strong><\/p>\n\n\n\n<p class=\"eplus-wrapper\">Let the length of the equal sides (base and height) be x.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">The length of the hypotenuse $= \\sqrt{2} \\times side = \\sqrt{2x}$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$\\sqrt{2x} = 4\\sqrt{2}$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Thus, $x = 4$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Hence, the length of the base and height is 4 inches .<\/p>\n\n\n\n<ol start=\"2\" class=\"eplus-wrapper wp-block-list\">\n<li class=\"eplus-wrapper\"><strong>One leg of the <\/strong>$45^\\circ\\;-\\;45^\\circ\\;-\\;90^\\circ$<strong> triangle is 5 feet. Find the length of the other sides of the triangle.<\/strong><\/li>\n<\/ol>\n\n\n\n<p class=\"eplus-wrapper\"><strong>Solution:&nbsp;<\/strong><\/p>\n\n\n\n<p class=\"eplus-wrapper\">The length of the leg of the triangle is 5 feet.&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper\">We know that the ratio of the sides equals $1 : 1 : \\sqrt{2}$. It means that two legs are congruent and the hypotenuse is $\\sqrt{2}$ times the length of the leg.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Thus, other leg $= 5$ feet<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Length of the hypotenuse $= \\sqrt{2} \\times leg$.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Hence, the length of the hypotenuse is $\\sqrt{2} \\times 5 = 5\\sqrt{2}$ feet.<\/p>\n\n\n\n<ol start=\"3\" class=\"eplus-wrapper wp-block-list\">\n<li class=\"eplus-wrapper\"><strong>&nbsp;The hypotenuse of the <\/strong>$45^\\circ\\;-\\;45^\\circ\\;-\\;90^\\circ$<strong> isosceles triangle is <\/strong>$8\\sqrt{2}$<strong> inches. Calculate the area of the triangle.<\/strong><\/li>\n<\/ol>\n\n\n\n<p class=\"eplus-wrapper\"><strong>Solution:<\/strong>&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper\">We know that<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Hypotenuse $= \\sqrt{2} \\times leg = \\sqrt{2x}$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$8\\sqrt{2} = \\sqrt{2x}$.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">We have leg $= x = 8$ inches&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Area of the triangle $=&nbsp;x \\times x \\times \\frac{1}{2} = \\frac{x^2}{2}$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Area of the triangle $= 8 \\times 8\\times \\frac{1}{2} = \\frac{8^2}{2} = \\frac{64}{2} = 32$ square inches.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">The area of the triangle is $32\\; inches^2$.<\/p>\n\n\n\n<ol start=\"4\" class=\"eplus-wrapper wp-block-list\">\n<li class=\"eplus-wrapper\"><strong>&nbsp;One of the legs of the <\/strong>$45^\\circ\\;-\\;45^\\circ\\;-\\;90^\\circ$<strong> triangle has a length of 29 feet. Calculate the perimeter of the triangle.<\/strong><\/li>\n<\/ol>\n\n\n\n<p class=\"eplus-wrapper\"><strong>Solution:&nbsp;<\/strong><\/p>\n\n\n\n<p class=\"eplus-wrapper\">One leg of the triangle $= 29$ feet<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Since two legs of a $45^\\circ\\;-\\;45^\\circ\\;-\\;90^\\circ$ triangle are congruent, we have&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper\">other leg $= 29$ feet<\/p>\n\n\n\n<p class=\"eplus-wrapper\">The length of the hypotenuse is $29\\sqrt{2}$ feet.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">To calculate the perimeter, we use the formula $x + x + \\sqrt{2x} = x(2 + \\sqrt{2})$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Hence, the perimeter would be $29 + 29 + 29\\sqrt{2} = 29(2 + \\sqrt{2})$.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">The perimeter of the triangle is $29(2 + \\sqrt{2})$ feet.<\/p>\n\n\n\n<ol start=\"5\" class=\"eplus-wrapper wp-block-list\">\n<li class=\"eplus-wrapper\"><strong>The shorter side of an isosceles right angled triangle is <\/strong>$9\\sqrt{2}$<strong> inches. Find the length of the diagonal of the triangle.<\/strong><\/li>\n<\/ol>\n\n\n\n<p class=\"eplus-wrapper\"><strong>Solution:<\/strong><\/p>\n\n\n\n<p class=\"eplus-wrapper\">The ratio of the sides of an isosceles right angled triangle is $x : x : \\sqrt{2x}$.&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper\">We have the given shorter side $x = 9\\sqrt{2}$ in<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Hence, the hypotenuse $= 9\\sqrt{2} \\times \\sqrt{2} = 9\\times2 = 18$ inches.<\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"10-practice-problems-on-45%C2%B0-45%C2%B0-90%C2%B0-triangle\">Practice Problems on 45\u00b0-45\u00b0-90\u00b0 Triangle<\/h2>\n\n\n\n<p class=\"eplus-wrapper\"><div class=\"spq_wrapper\"><h2 style=\"display:none;\">45\u00b0-45\u00b0-90\u00b0 Triangle: Properties, Formulas, Construction<\/h2><p style=\"display:none;\">Attend this quiz & Test your knowledge.<\/p><div class=\"spq_question_wrapper\" data-answer=\"1\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">1<\/span><h3 class=\"sqp_question_text\">What is the ratio of the two legs of a $45^\\circ\\;-\\;45^\\circ\\;-\\;90^\\circ$ triangle?<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">1:2<\/div><div class=\"spq_answer_block\" data-value=\"1\">1:1<\/div><div class=\"spq_answer_block\" data-value=\"2\">$1:\\sqrt{2}$<\/div><div class=\"spq_answer_block\" data-value=\"3\">1:3<\/div><div class=\"sqp_question_hint\"><div class=\"sqp_question_hint__header\"><span class=\"spq_correct\">Correct<\/span><span class=\"spq_incorrect\">Incorrect<\/span><\/div><div class=\"sqp_question_hint__content\"><span>Correct answer is: 1:1<br\/>Two legs of a $45^\\circ\\;-\\;45^\\circ\\;-\\;90^\\circ$ triangle are equal. Thus, their ratio will be 1:1.<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"2\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">2<\/span><h3 class=\"sqp_question_text\">Two $45^\\circ\\;-\\;45^\\circ\\;-\\;90^\\circ$ triangles joined together along the hypotenuse form a<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">rectangle<\/div><div class=\"spq_answer_block\" data-value=\"1\">pentagon<\/div><div class=\"spq_answer_block\" data-value=\"2\">square<\/div><div class=\"spq_answer_block\" data-value=\"3\">triangle<\/div><div class=\"sqp_question_hint\"><div class=\"sqp_question_hint__header\"><span class=\"spq_correct\">Correct<\/span><span class=\"spq_incorrect\">Incorrect<\/span><\/div><div class=\"sqp_question_hint__content\"><span>Correct answer is: square<br\/>A $45^\\circ\\;-\\;45^\\circ\\;-\\;90^\\circ$ triangle is half of a square. In other words, if you join two $45^\\circ\\;-\\;45^\\circ\\;-\\;90^\\circ$ triangles along the hypotenuse, you get a square with side length equal to the length of the legs.<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"3\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">3<\/span><h3 class=\"sqp_question_text\">If the length of the shortest side of an isosceles right-angled triangle is 3 feet, what is the length of the hypotenuse<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">3 feet<\/div><div class=\"spq_answer_block\" data-value=\"1\">$6\\sqrt{2}$ feet<\/div><div class=\"spq_answer_block\" data-value=\"2\">$9\\sqrt{2}$ feet<\/div><div class=\"spq_answer_block\" data-value=\"3\">$3\\sqrt{2}$ feet<\/div><div class=\"sqp_question_hint\"><div class=\"sqp_question_hint__header\"><span class=\"spq_correct\">Correct<\/span><span class=\"spq_incorrect\">Incorrect<\/span><\/div><div class=\"sqp_question_hint__content\"><span>Correct answer is: $3\\sqrt{2}$ feet<br\/>The length of the hypotenuse of this triangle is equal to 2 times the leg of the triangle.<br>\r\nHypotenuse $= 3\\sqrt{2}$ feet<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"3\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">4<\/span><h3 class=\"sqp_question_text\">What is the perimeter of a $45^\\circ\\;-\\;45^\\circ\\;-\\;90^\\circ$ triangle with leg a?<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">3a<\/div><div class=\"spq_answer_block\" data-value=\"1\">$3\\sqrt{2a}$<\/div><div class=\"spq_answer_block\" data-value=\"2\">$6(a + \\sqrt{2})$<\/div><div class=\"spq_answer_block\" data-value=\"3\">$a(2 + \\sqrt{2})$<\/div><div class=\"sqp_question_hint\"><div class=\"sqp_question_hint__header\"><span class=\"spq_correct\">Correct<\/span><span class=\"spq_incorrect\">Incorrect<\/span><\/div><div class=\"sqp_question_hint__content\"><span>Correct answer is: $a(2 + \\sqrt{2})$<br\/>The perimeter of the triangle with leg a  is $a + a + \\sqrt{2a} = a(2 + \\sqrt{2})$ units<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"2\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">5<\/span><h3 class=\"sqp_question_text\">What is the area of the $45^\\circ\\;-\\;45^\\circ\\;-\\;90^\\circ$ triangle with hypotenuse $2\\sqrt{2}$ inches?<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">4 inches<\/div><div class=\"spq_answer_block\" data-value=\"1\">8 inches<\/div><div class=\"spq_answer_block\" data-value=\"2\">2 inches<\/div><div class=\"spq_answer_block\" data-value=\"3\">6 inches<\/div><div class=\"sqp_question_hint\"><div class=\"sqp_question_hint__header\"><span class=\"spq_correct\">Correct<\/span><span class=\"spq_incorrect\">Incorrect<\/span><\/div><div class=\"sqp_question_hint__content\"><span>Correct answer is: 2 inches<br\/>Hypotenuse $= \\sqrt{2} \\times leg = 2\\sqrt{2}$ inches<br>\r\nThus, length of the legs $= 2$ inches<br>\r\nBase = Height$ = 2$ in<br>\r\nThe area of the given triangle $= \\frac{1}{2} \\times$ Base $\\times$ Height $= \\frac{1}{2} \\times 2 \\times 2 = 2\\; inches^2$.<\/span><\/div><\/div><\/div><\/div>  <script type=\"application\/ld+json\">{\n        \"@context\": \"https:\/\/schema.org\/\", \n        \"@type\": \"Quiz\", \n        \"typicalAgeRange\": \"3-11\",\n        \"educationalLevel\":  \"beginner\",\n        \"assesses\" : \"Attend this quiz & Test your knowledge.\",\n        \"educationalAlignment\": [\n              {\n                \"@type\": \"AlignmentObject\",\n                \"alignmentType\": \"educationalSubject\",\n                \"targetName\": \"Math\"\n              }] ,\n        \"name\": \"45\u00b0-45\u00b0-90\u00b0 Triangle: Properties, Formulas, Construction\",        \n        \"about\": {\n                \"@type\": \"Thing\",\n                \"name\": \"45\u00b0-45\u00b0-90\u00b0 Triangle\"\n        },  \n        \"hasPart\": [{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"What is the ratio of the two legs of a $$45^\\\\circ\\\\;-\\\\;45^\\\\circ\\\\;-\\\\;90^\\\\circ$$ triangle?\",\n                    \"text\": \"What is the ratio of the two legs of a $$45^\\\\circ\\\\;-\\\\;45^\\\\circ\\\\;-\\\\;90^\\\\circ$$ triangle?\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"Two legs of a $$45^\\\\circ\\\\;-\\\\;45^\\\\circ\\\\;-\\\\;90^\\\\circ$$ triangle are equal. 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Thus, their ratio will be 1:1.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 2,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$1:\\\\sqrt{2}$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"Two legs of a $$45^\\\\circ\\\\;-\\\\;45^\\\\circ\\\\;-\\\\;90^\\\\circ$$ triangle are equal. Thus, their ratio will be 1:1.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 3,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"1:3\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"Two legs of a $$45^\\\\circ\\\\;-\\\\;45^\\\\circ\\\\;-\\\\;90^\\\\circ$$ triangle are equal. Thus, their ratio will be 1:1.\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 1,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"1:1\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"Two legs of a $$45^\\\\circ\\\\;-\\\\;45^\\\\circ\\\\;-\\\\;90^\\\\circ$$ triangle are equal. Thus, their ratio will be 1:1.\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"Two legs of a $$45^\\\\circ\\\\;-\\\\;45^\\\\circ\\\\;-\\\\;90^\\\\circ$$ triangle are equal. Thus, their ratio will be 1:1.\"\n                      }\n                    } \n\n                    },{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"Two $$45^\\\\circ\\\\;-\\\\;45^\\\\circ\\\\;-\\\\;90^\\\\circ$$ triangles joined together along the hypotenuse form a\",\n                    \"text\": \"Two $$45^\\\\circ\\\\;-\\\\;45^\\\\circ\\\\;-\\\\;90^\\\\circ$$ triangles joined together along the hypotenuse form a\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"A $$45^\\\\circ\\\\;-\\\\;45^\\\\circ\\\\;-\\\\;90^\\\\circ$$ triangle is half of a square. In other words, if you join two $$45^\\\\circ\\\\;-\\\\;45^\\\\circ\\\\;-\\\\;90^\\\\circ$$ triangles along the hypotenuse, you get a square with side length equal to the length of the legs.\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 0,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"rectangle\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"A $$45^\\\\circ\\\\;-\\\\;45^\\\\circ\\\\;-\\\\;90^\\\\circ$$ triangle is half of a square. In other words, if you join two $$45^\\\\circ\\\\;-\\\\;45^\\\\circ\\\\;-\\\\;90^\\\\circ$$ triangles along the hypotenuse, you get a square with side length equal to the length of the legs.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 1,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"pentagon\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"A $$45^\\\\circ\\\\;-\\\\;45^\\\\circ\\\\;-\\\\;90^\\\\circ$$ triangle is half of a square. In other words, if you join two $$45^\\\\circ\\\\;-\\\\;45^\\\\circ\\\\;-\\\\;90^\\\\circ$$ triangles along the hypotenuse, you get a square with side length equal to the length of the legs.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 3,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"triangle\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"A $$45^\\\\circ\\\\;-\\\\;45^\\\\circ\\\\;-\\\\;90^\\\\circ$$ triangle is half of a square. In other words, if you join two $$45^\\\\circ\\\\;-\\\\;45^\\\\circ\\\\;-\\\\;90^\\\\circ$$ triangles along the hypotenuse, you get a square with side length equal to the length of the legs.\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 2,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"square\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"A $$45^\\\\circ\\\\;-\\\\;45^\\\\circ\\\\;-\\\\;90^\\\\circ$$ triangle is half of a square. In other words, if you join two $$45^\\\\circ\\\\;-\\\\;45^\\\\circ\\\\;-\\\\;90^\\\\circ$$ triangles along the hypotenuse, you get a square with side length equal to the length of the legs.\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"A $$45^\\\\circ\\\\;-\\\\;45^\\\\circ\\\\;-\\\\;90^\\\\circ$$ triangle is half of a square. In other words, if you join two $$45^\\\\circ\\\\;-\\\\;45^\\\\circ\\\\;-\\\\;90^\\\\circ$$ triangles along the hypotenuse, you get a square with side length equal to the length of the legs.\"\n                      }\n                    } \n\n                    },{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"If the length of the shortest side of an isosceles right-angled triangle is 3 feet, what is the length of the hypotenuse\",\n                    \"text\": \"If the length of the shortest side of an isosceles right-angled triangle is 3 feet, what is the length of the hypotenuse\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"The length of the hypotenuse of this triangle is equal to 2 times the leg of the triangle.<br>\r\nHypotenuse $$= 3\\\\sqrt{2}$$ feet\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 0,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"3 feet\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"The length of the hypotenuse of this triangle is equal to 2 times the leg of the triangle.<br>\r\nHypotenuse $$= 3\\\\sqrt{2}$$ feet\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 1,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$6\\\\sqrt{2}$$ feet\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"The length of the hypotenuse of this triangle is equal to 2 times the leg of the triangle.<br>\r\nHypotenuse $$= 3\\\\sqrt{2}$$ feet\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 2,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$9\\\\sqrt{2}$$ feet\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"The length of the hypotenuse of this triangle is equal to 2 times the leg of the triangle.<br>\r\nHypotenuse $$= 3\\\\sqrt{2}$$ feet\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 3,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"$$3\\\\sqrt{2}$$ feet\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"The length of the hypotenuse of this triangle is equal to 2 times the leg of the triangle.<br>\r\nHypotenuse $$= 3\\\\sqrt{2}$$ feet\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"The length of the hypotenuse of this triangle is equal to 2 times the leg of the triangle.<br>\r\nHypotenuse $$= 3\\\\sqrt{2}$$ feet\"\n                      }\n                    } \n\n                    },{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"What is the perimeter of a $$45^\\\\circ\\\\;-\\\\;45^\\\\circ\\\\;-\\\\;90^\\\\circ$$ triangle with leg a?\",\n                    \"text\": \"What is the perimeter of a $$45^\\\\circ\\\\;-\\\\;45^\\\\circ\\\\;-\\\\;90^\\\\circ$$ triangle with leg a?\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"The perimeter of the triangle with leg a  is $$a + a + \\\\sqrt{2a} = a(2 + \\\\sqrt{2})$$ units\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 0,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"3a\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"The perimeter of the triangle with leg a  is $$a + a + \\\\sqrt{2a} = a(2 + \\\\sqrt{2})$$ units\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 1,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$3\\\\sqrt{2a}$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"The perimeter of the triangle with leg a  is $$a + a + \\\\sqrt{2a} = a(2 + \\\\sqrt{2})$$ units\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 2,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$6(a + \\\\sqrt{2})$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"The perimeter of the triangle with leg a  is $$a + a + \\\\sqrt{2a} = a(2 + \\\\sqrt{2})$$ units\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 3,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"$$a(2 + \\\\sqrt{2})$$\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"The perimeter of the triangle with leg a  is $$a + a + \\\\sqrt{2a} = a(2 + \\\\sqrt{2})$$ units\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"The perimeter of the triangle with leg a  is $$a + a + \\\\sqrt{2a} = a(2 + \\\\sqrt{2})$$ units\"\n                      }\n                    } \n\n                    },{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"What is the area of the $$45^\\\\circ\\\\;-\\\\;45^\\\\circ\\\\;-\\\\;90^\\\\circ$$ triangle with hypotenuse $$2\\\\sqrt{2}$$ inches?\",\n                    \"text\": \"What is the area of the $$45^\\\\circ\\\\;-\\\\;45^\\\\circ\\\\;-\\\\;90^\\\\circ$$ triangle with hypotenuse $$2\\\\sqrt{2}$$ inches?\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"Hypotenuse $$= \\\\sqrt{2} \\\\times leg = 2\\\\sqrt{2}$$ inches<br>\r\nThus, length of the legs $$= 2$$ inches<br>\r\nBase = Height$$ = 2$$ in<br>\r\nThe area of the given triangle $$= \\\\frac{1}{2} \\\\times$$ Base $$\\\\times$$ Height $$= \\\\frac{1}{2} \\\\times 2 \\\\times 2 = 2\\\\; inches^2$$.\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 0,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"4 inches\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"Hypotenuse $$= \\\\sqrt{2} \\\\times leg = 2\\\\sqrt{2}$$ inches<br>\r\nThus, length of the legs $$= 2$$ inches<br>\r\nBase = Height$$ = 2$$ in<br>\r\nThe area of the given triangle $$= \\\\frac{1}{2} \\\\times$$ Base $$\\\\times$$ Height $$= \\\\frac{1}{2} \\\\times 2 \\\\times 2 = 2\\\\; inches^2$$.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 1,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"8 inches\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"Hypotenuse $$= \\\\sqrt{2} \\\\times leg = 2\\\\sqrt{2}$$ inches<br>\r\nThus, length of the legs $$= 2$$ inches<br>\r\nBase = Height$$ = 2$$ in<br>\r\nThe area of the given triangle $$= \\\\frac{1}{2} \\\\times$$ Base $$\\\\times$$ Height $$= \\\\frac{1}{2} \\\\times 2 \\\\times 2 = 2\\\\; inches^2$$.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 3,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"6 inches\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"Hypotenuse $$= \\\\sqrt{2} \\\\times leg = 2\\\\sqrt{2}$$ inches<br>\r\nThus, length of the legs $$= 2$$ inches<br>\r\nBase = Height$$ = 2$$ in<br>\r\nThe area of the given triangle $$= \\\\frac{1}{2} \\\\times$$ Base $$\\\\times$$ Height $$= \\\\frac{1}{2} \\\\times 2 \\\\times 2 = 2\\\\; inches^2$$.\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 2,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"2 inches\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"Hypotenuse $$= \\\\sqrt{2} \\\\times leg = 2\\\\sqrt{2}$$ inches<br>\r\nThus, length of the legs $$= 2$$ inches<br>\r\nBase = Height$$ = 2$$ in<br>\r\nThe area of the given triangle $$= \\\\frac{1}{2} \\\\times$$ Base $$\\\\times$$ Height $$= \\\\frac{1}{2} \\\\times 2 \\\\times 2 = 2\\\\; inches^2$$.\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"Hypotenuse $$= \\\\sqrt{2} \\\\times leg = 2\\\\sqrt{2}$$ inches<br>\r\nThus, length of the legs $$= 2$$ inches<br>\r\nBase = Height$$ = 2$$ in<br>\r\nThe area of the given triangle $$= \\\\frac{1}{2} \\\\times$$ Base $$\\\\times$$ Height $$= \\\\frac{1}{2} \\\\times 2 \\\\times 2 = 2\\\\; inches^2$$.\"\n                      }\n                    } \n\n                    }]}<\/script><\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"11-frequently-asked-questions-on-45%C2%B0-45%C2%B0-90%C2%B0-triangle\">Frequently Asked Questions on 45\u00b0-45\u00b0-90\u00b0 Triangle<\/h2>\n\n\n<div class=\"wp-block-ub-content-toggle\" id=\"ub-content-toggle-f07661b5-33e1-4576-bc05-1e6566f6e51c\" data-mobilecollapse=\"true\" data-desktopcollapse=\"true\">\n<div class=\"wp-block-ub-content-toggle-accordion\">\n                <div class=\"wp-block-ub-content-toggle-accordion-title-wrap\" aria-expanded=\"false\" aria-controls=\"ub-content-toggle-panel-0-f07661b5-33e1-4576-bc05-1e6566f6e51c\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-f07661b5-33e1-4576-bc05-1e6566f6e51c\"><strong>What is the use of the ratio of sides in the <\/strong>$45^\\circ\\;-\\;45^\\circ\\;-\\;90^\\circ$<strong> triangle?<\/strong><\/p><div class=\"wp-block-ub-content-toggle-accordion-toggle-wrap right\"><span class=\"wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-chevron-down\"><\/span>\n                    <\/div><\/div><div role=\"region\" class=\"wp-block-ub-content-toggle-accordion-content-wrap ub-hide\" id=\"ub-content-toggle-panel-0-f07661b5-33e1-4576-bc05-1e6566f6e51c\">\n\n<p class=\"eplus-wrapper\">We can find the length of the missing side using the ratio of the sides.&nbsp; The ratio of the sides equals $x : x : \\sqrt{2x}$. If you know the hypotenuse, you can find the length of the legs by dividing it by $\\sqrt{2}$. If you know the length of the shorter side\/leg, you can find the length of the hypotenuse by multiplying the side by $\\sqrt{2}$.<\/p>\n\n<\/div><\/div>\n\n<div class=\"wp-block-ub-content-toggle-accordion\">\n                <div class=\"wp-block-ub-content-toggle-accordion-title-wrap\" aria-expanded=\"false\" aria-controls=\"ub-content-toggle-panel-1-f07661b5-33e1-4576-bc05-1e6566f6e51c\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-f07661b5-33e1-4576-bc05-1e6566f6e51c\"><strong>What is a <\/strong>$30^\\circ\\;-\\;60^\\circ\\;-\\;90^\\circ$<strong> triangle?<\/strong><\/p><div class=\"wp-block-ub-content-toggle-accordion-toggle-wrap right\"><span class=\"wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-chevron-down\"><\/span>\n                    <\/div><\/div><div role=\"region\" class=\"wp-block-ub-content-toggle-accordion-content-wrap ub-hide\" id=\"ub-content-toggle-panel-1-f07661b5-33e1-4576-bc05-1e6566f6e51c\">\n\n<p class=\"eplus-wrapper\">A $30^\\circ\\;-\\;60^\\circ\\;-\\;90^\\circ$ triangle is a triangle in which there is one $90^\\circ$ angle, one $30^\\circ$ degree angle, and one $60^\\circ$ angle.<\/p>\n\n<\/div><\/div>\n\n<div class=\"wp-block-ub-content-toggle-accordion\">\n                <div class=\"wp-block-ub-content-toggle-accordion-title-wrap\" aria-expanded=\"false\" aria-controls=\"ub-content-toggle-panel-2-f07661b5-33e1-4576-bc05-1e6566f6e51c\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-f07661b5-33e1-4576-bc05-1e6566f6e51c\"><strong>Why are all <\/strong>$45^\\circ\\;-\\;45^\\circ\\;-\\;90^\\circ$<strong> triangles similar?<\/strong><\/p><div class=\"wp-block-ub-content-toggle-accordion-toggle-wrap right\"><span class=\"wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-chevron-down\"><\/span>\n                    <\/div><\/div><div role=\"region\" class=\"wp-block-ub-content-toggle-accordion-content-wrap ub-hide\" id=\"ub-content-toggle-panel-2-f07661b5-33e1-4576-bc05-1e6566f6e51c\">\n\n<p class=\"eplus-wrapper\">For all $45^\\circ\\;-\\;45^\\circ\\;-\\;90^\\circ$ triangles, the measures of the three interior angles is always the same. Thus, all such triangles may have different sizes but always have the same shape. By AA similarity criterion, all $45^\\circ\\;-\\;45^\\circ\\;-\\;90^\\circ$ triangles are similar.<\/p>\n\n<\/div><\/div>\n\n<div class=\"wp-block-ub-content-toggle-accordion\">\n                <div class=\"wp-block-ub-content-toggle-accordion-title-wrap\" aria-expanded=\"false\" aria-controls=\"ub-content-toggle-panel-3-f07661b5-33e1-4576-bc05-1e6566f6e51c\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-f07661b5-33e1-4576-bc05-1e6566f6e51c\"><strong>Is a <\/strong>$45^\\circ\\;-\\;45^\\circ\\;-\\;90^\\circ$<strong> degree triangle isosceles?<\/strong><\/p><div class=\"wp-block-ub-content-toggle-accordion-toggle-wrap right\"><span class=\"wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-chevron-down\"><\/span>\n                    <\/div><\/div><div role=\"region\" class=\"wp-block-ub-content-toggle-accordion-content-wrap ub-hide\" id=\"ub-content-toggle-panel-3-f07661b5-33e1-4576-bc05-1e6566f6e51c\">\n\n<p class=\"eplus-wrapper\">Yes, a $45^\\circ\\;-\\;45^\\circ\\;-\\;90^\\circ$ degree triangle is a right isosceles triangle.<\/p>\n\n<\/div><\/div>\n<\/div>","protected":false},"excerpt":{"rendered":"<p>What Is a 45\u00b0-45\u00b0-90\u00b0 Triangle? The $45^\\circ\\;-\\;45^\\circ\\;-\\;90^\\circ$ triangle, as the name itself suggests, is a triangle in which there is one $90^\\circ$ angle (right angle), and two $45^\\circ$ angles.&nbsp; Since two angles are congruent, the sides opposite to the congruent angles are also congruent. Two congruent sides and one right angle make the $45^\\circ\\;-\\;45^\\circ\\;-\\;90^\\circ$ triangle &#8230; <a title=\"45\u00b0-45\u00b0-90\u00b0 Triangle: Properties, Formulas, Construction\" class=\"read-more\" href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/45-45-90-triangle\" aria-label=\"More on 45\u00b0-45\u00b0-90\u00b0 Triangle: Properties, Formulas, Construction\">Read more<\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"ub_ctt_via":"","footnotes":""},"categories":[1],"tags":[],"class_list":["post-29365","post","type-post","status-publish","format-standard","hentry","category-all"],"featured_image_src":null,"author_info":{"display_name":"Mithun Jhawar","author_link":"https:\/\/www.splashlearn.com\/math-vocabulary\/author\/mithun-jhawarsplashlearn-com\/"},"acf":[],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts\/29365","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/comments?post=29365"}],"version-history":[{"count":15,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts\/29365\/revisions"}],"predecessor-version":[{"id":37397,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts\/29365\/revisions\/37397"}],"wp:attachment":[{"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/media?parent=29365"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/categories?post=29365"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/tags?post=29365"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}