{"id":23497,"date":"2023-02-03T04:30:43","date_gmt":"2023-02-03T04:30:43","guid":{"rendered":"https:\/\/www.splashlearn.com\/math-vocabulary\/?page_id=23497"},"modified":"2024-01-16T10:19:48","modified_gmt":"2024-01-16T10:19:48","slug":"pythagorean-triples","status":"publish","type":"post","link":"https:\/\/www.splashlearn.com\/math-vocabulary\/pythagorean-triples","title":{"rendered":"Pythagorean Triples"},"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-207ab041-c804-4839-9bca-0ea14c083eec\" 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\/pythagorean-triples#0-introduction-to-pythagorean-triples>Introduction to Pythagorean Triples<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/pythagorean-triples#2-pythagorean-triples-definition>Pythagorean Triples: Definition<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/pythagorean-triples#4-pythagorean-triples-list>Pythagorean Triples List<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/pythagorean-triples#14-solved-examples>Solved Examples<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/pythagorean-triples#15-practice-problems>Practice Problems<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/pythagorean-triples#16-frequently-asked-questions>Frequently Asked Questions<\/a><\/li><\/ul><\/div><\/div><\/div>\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"0-introduction-to-pythagorean-triples\">Introduction to Pythagorean Triples<\/h2>\n\n\n\n<p class=\"eplus-wrapper\">Pythagorean triples are three positive integers which satisfy the Pythagoras\u2019 theorem. Pythagoras\u2019 theorem states that in any right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides of the right triangle.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full eplus-wrapper\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"428\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/02\/Pythagorean-Triples-1.png\" alt=\"Pythagorean triples and Pythagoras\u2019 theorem\" class=\"wp-image-23592\" title=\"Pythagorean triples and Pythagoras\u2019 theorem\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/02\/Pythagorean-Triples-1.png 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/02\/Pythagorean-Triples-1-300x207.png 300w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n\n\n\n<p class=\"eplus-wrapper\">Any three positive integers that fully satisfy the Pythagorean theorem are known as \u201cPythagorean triples.\u201d&nbsp;<\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"1-what-are-pythagorean-triples-in-geometry\">What Are Pythagorean Triples in Geometry?<\/h2>\n\n\n\n<p class=\"eplus-wrapper\">Pythagorean triples, in simple words, are the integer solutions to the Pythagoras\u2019 theorem, containing positive integers.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Here, \u201cc\u201d is the \u201chypotenuse\u201d or the longest side of the triangle, and \u201ca\u201d and \u201cb\u201d are the other two sides of the right-angled triangle. We represent the Pythagorean triples as $(a, b, c)$.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">The three sides a, b, c of the any right angled triangle satisfy $a^{2} + b^{2}  = c^{2}$.&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper\">So, $(a, b, c)$ form a Pythagorean triple.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full eplus-wrapper\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"419\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/02\/Pythagorean-Triples-2.png\" alt=\"Pythagorean triples follow Pythagorean theorem\" class=\"wp-image-23593\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/02\/Pythagorean-Triples-2.png 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/02\/Pythagorean-Triples-2-300x203.png 300w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"2-pythagorean-triples-definition\">Pythagorean Triples: Definition<\/h2>\n\n\n\n<p class=\"eplus-wrapper\">A set of 3 positive numbers that satisfy the formula of the Pythagoras\u2019 theorem that is expressed as $a^{2} + b^{2} = c^{2}$, where a, b, and c are positive integers, are called Pythagorean triples.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Example: $(3, 4, 5)$ is the first known, the smallest and the most popular example of Pythagorean triple.&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Let us verify if 3, 4 and 5 satisfy the equation $a^{2} + b^{2} = c^{2}$.&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$3^{2} + 4^{2} = 9 + 16 = 25 = 5^{2}$<\/p>\n\n\n\n<p class=\"eplus-wrapper\"><strong>What is the Formula for Pythagorean Triples?<\/strong><\/p>\n\n\n\n<p class=\"eplus-wrapper\">For a right-angled triangle, there exists a relationship between the three sides of the triangle.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">If the longest side is called the hypotenuse, the square of the hypotenuse is equal to the sum of the squares of the other two sides.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">So, we can form the formula for the Pythagoras triples as&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$a^{2} + b^{2} = h^{2}$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">where&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper\">h is the hypotenuse,<\/p>\n\n\n\n<p class=\"eplus-wrapper\">a and b are the other two sides.<\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"3-pythagorean-triples-examples\">Pythagorean Triples: Examples<\/h2>\n\n\n\n<p class=\"eplus-wrapper\">There is an endless list of Pythagorean triples. $(3, 4, 5)$ is the first Pythagorean triple. A few more triples can be generated by multiplying all the original numbers by another positive integer.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full eplus-wrapper\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"291\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/02\/Pythagorean-Triples-3.png\" alt=\"Pythagorean triples generated using 3-4-5 triangle list\" class=\"wp-image-23595\" title=\"Pythagorean triples generated using 3-4-5 triangle list\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/02\/Pythagorean-Triples-3.png 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/02\/Pythagorean-Triples-3-300x141.png 300w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"4-pythagorean-triples-list\">Pythagorean Triples List<\/h2>\n\n\n\n<p class=\"eplus-wrapper\">It is not possible to list down all Pythagorean triples since the list is endless. The table given below shows some examples of Pythagorean triples.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full eplus-wrapper\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"477\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/02\/Pythagorean-Triples-4.png\" alt=\"Pythagorean triples list\" class=\"wp-image-23596\" title=\"Pythagorean triples list\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/02\/Pythagorean-Triples-4.png 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/02\/Pythagorean-Triples-4-300x231.png 300w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"5-common-pythagorean-triples\">Common Pythagorean Triples<\/h2>\n\n\n\n<p class=\"eplus-wrapper\">Here are a few common Pythagorean triples.<\/p>\n\n\n\n<ul class=\"eplus-wrapper wp-block-list\">\n<li class=\"eplus-wrapper\">3-4-5<\/li>\n\n\n\n<li class=\"eplus-wrapper\">5-12-13<\/li>\n\n\n\n<li class=\"eplus-wrapper\">6-8-10<\/li>\n\n\n\n<li class=\"eplus-wrapper\">9-12-15<\/li>\n\n\n\n<li class=\"eplus-wrapper\">12-16-20<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"6-what-are-primitive-pythagorean-triples\">What are Primitive Pythagorean Triples?<\/h2>\n\n\n\n<p class=\"eplus-wrapper\">The set of three numbers which have no common divisor other than 1 is considered to be a primitive Pythagorean triple.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">The first Pythagorean triple is primitive, since 3, 4, and 5 have no common divisors other than 1. An example of an imprimitive or non-primitive Pythagorean triple is $(15, 20, 25)$ multiple of 5.<\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"7-how-to-find-pythagorean-triples\">How to Find Pythagorean Triples?<\/h2>\n\n\n\n<p class=\"eplus-wrapper\">Let\u2019s understand how to find Pythagorean triples using an example.<\/p>\n\n\n\n<ol class=\"eplus-wrapper wp-block-list\">\n<li class=\"eplus-wrapper\">Pick an even number (other than 2) to be the longer leg\u2019s length.<\/li>\n<\/ol>\n\n\n\n<p class=\"eplus-wrapper\">So, let $b = 4$<\/p>\n\n\n\n<ol start=\"2\" class=\"eplus-wrapper wp-block-list\">\n<li class=\"eplus-wrapper\">Find a prime number one greater than that even number, to be the hypotenuse.&nbsp;<\/li>\n<\/ol>\n\n\n\n<p class=\"eplus-wrapper\">So, hypotenuse $= h = 4 + 1 = 5$<\/p>\n\n\n\n<ol start=\"3\" class=\"eplus-wrapper wp-block-list\">\n<li class=\"eplus-wrapper\">Calculate the third value to find the Pythagorean triple.<\/li>\n<\/ol>\n\n\n\n<p class=\"eplus-wrapper\">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;We know that, $a^{2} + b^{2}&nbsp; = h^{2}$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Thus,&nbsp; $a^{2} + 4^{2} = 5^{2}$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$a^{2} = 25 \u2212 16 = 9$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$a = 3$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Pythagorean triple $= (3, 4, 5)$<\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"8-formula-for-generating-pythagorean-triples\">Formula for Generating Pythagorean Triples<\/h2>\n\n\n\n<p class=\"eplus-wrapper\">To find the Pythagorean triples, the following formula can also be used. If a, b are two sides of the right triangle and c is the hypotenuse, then<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$a = m^{2} \u2212 n^{2}$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$b = 2 mn$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$c = m^{2} + n^{2}$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">These values form sides of a right triangle or Pythagorean triple.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">(i) m, n are positive integers.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">(ii) $m &gt; n$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">(iii) m and n are coprime and both should not be odd numbers.<\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"9-how-to-form-pythagorean-triples\">How to Form Pythagorean Triples?<\/h2>\n\n\n\n<p class=\"eplus-wrapper\">Let us now see how we can find or generate Pythagorean triples using formulas.<\/p>\n\n\n\n<p class=\"eplus-wrapper\"><strong>Case 1: For odd numbers<\/strong><\/p>\n\n\n\n<p class=\"eplus-wrapper\">Let \u2018x\u2019 be an odd number.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">The Pythagorean triple will be:&nbsp;<\/p>\n\n\n\n<ol class=\"eplus-wrapper wp-block-list\">\n<li class=\"eplus-wrapper\">$x$<\/li>\n\n\n\n<li class=\"eplus-wrapper\">$\\frac{x^{2}}{2} \u2212 0.5$<\/li>\n\n\n\n<li class=\"eplus-wrapper\">$\\frac{x^{2}}{2} + 0.5$.<\/li>\n<\/ol>\n\n\n\n<p class=\"eplus-wrapper\">Let us take an example: (3, 4, 5). Now, let us see how to form this Pythagorean triple.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Here,&nbsp; $x = 3$, which is an odd number.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$\\frac{x^{2}}{2} \u2212 0.5 = \\frac{9}{2} \u2212 0.5 = 4.5 \u2212 0.5 = 4$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$\\frac{x^{2}}{2} + 0.5 = \\frac{9}{2} + 0.5 = 4.5 + 0.5 = 5$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Hence, the Pythagorean triple formed is $(3, 4, 5)$.<\/p>\n\n\n\n<p class=\"eplus-wrapper\"><strong>Case 2: For even numbers<\/strong><\/p>\n\n\n\n<p class=\"eplus-wrapper\">If \u201cx\u201d is an even number, then the Pythagorean triple is given as<\/p>\n\n\n\n<ol class=\"eplus-wrapper wp-block-list\">\n<li class=\"eplus-wrapper\">$x^{2}$,&nbsp;<\/li>\n\n\n\n<li class=\"eplus-wrapper\">$\\frac{x^{2}}{2} \u2212 1$,&nbsp;<\/li>\n\n\n\n<li class=\"eplus-wrapper\">$\\frac{x^{2}}{2} + 1$.<\/li>\n<\/ol>\n\n\n\n<p class=\"eplus-wrapper\">Let us take an example, (16, 63, 65). Now, we will check how to form the Pythagorean triple.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Here, $x = 16$, which is an even number.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$\\frac{x^{2}}{2} \u2212 1 = \\frac{16^{2}}{2} \u2212 1 = 82 \u2212 1 = 64 \u2212 1 = 63$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$\\frac{x^{2}}{2} +1 = \\frac{16^{2}}{2} + 1 = 82 + 1 = 64 + 1 = 65$.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Hence, we get the Pythagorean triple $(16, 63, 65)$<\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"10-pythagorean-triples-formula-proof\">Pythagorean Triples Formula: Proof<\/h2>\n\n\n\n<p class=\"eplus-wrapper\">Consider a square shown below with side (a+b).<\/p>\n\n\n\n<figure class=\"wp-block-image size-full eplus-wrapper\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"395\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/02\/Pythagorean-Triples-5.png\" alt=\"Pythagorean triples proof\" class=\"wp-image-23597\" title=\"Pythagorean triples proof\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/02\/Pythagorean-Triples-5.png 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/02\/Pythagorean-Triples-5-300x191.png 300w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n\n\n\n<p class=\"eplus-wrapper\">Area of the given square $= (a + b)^{2}$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Area of each small right triangle $= \\frac{1}{2} ab$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Area of 4 right triangles $= 4 \\times \\frac{1}{2} ab = 2ab$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Area of the yellow square formed inside $= side^{2} = c^{2}$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Here, area of the whole square $=$ Area of 4 right triangles $+$ Area of Yellow square<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$(a + b)^{2} = 2ab + c^{2}$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$a^{2} + 2ab + b^{2} =&nbsp;2ab+ c^{2}$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$a^{2} + b^{2} =  c^{2}$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Hence proved.<\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"11-what-are-triangular-numbers\">What are Triangular Numbers?<\/h2>\n\n\n\n<p class=\"eplus-wrapper\">A triangular number or triangle number is used to count objects arranged in an equilateral triangle. A triangular number is a number that can be written as the sum of the first n positive integers.&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-image size-full eplus-wrapper\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"758\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/02\/Pythagorean-Triples-6.png\" alt=\"Triangular numbers\" class=\"wp-image-23598\" title=\"Triangular numbers\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/02\/Pythagorean-Triples-6.png 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/02\/Pythagorean-Triples-6-245x300.png 245w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n\n\n\n<p class=\"eplus-wrapper\">The $n^{th}$ triangular number is the sum of the n natural numbers from 1 to n. The sequence of triangular numbers, starting with the 0th triangular number, is&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper\">0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66 and so on.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Each perfect square is the sum of two successive triangular numbers.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$0 + 1 = 1$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$1 + 3 = 4$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$3 + 6 = 9$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$6 + 10 = 16$; and so on.<\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"12-fun-facts\">Fun Facts!<\/h2>\n\n\n\n<ul class=\"eplus-wrapper wp-block-list\">\n<li class=\"eplus-wrapper\">The 5 most common Pythagorean triples are $(3, 4, 5), (5, 12, 13), (6, 8, 10), (9, 12, 15)$, and $(15, 20, 25)$.<\/li>\n\n\n\n<li class=\"eplus-wrapper\">Pythagorean triples contain either all even numbers or two odd numbers and an even number.&nbsp;<\/li>\n\n\n\n<li class=\"eplus-wrapper\">A Pythagorean triple never be made up of all odd numbers or two even numbers and an odd number.<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"13-conclusion\">Conclusion<\/h2>\n\n\n\n<p class=\"eplus-wrapper\">In this article, we learned what Pythagorean triples are. We saw many examples of Pythagorean triples and learnt the formula to find them. We also learned about triangular numbers and saw some facts about Pythagorean triples. Now let us solve some practice problems to understand the above concepts well.&nbsp;<\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"14-solved-examples\">Solved Examples<\/h2>\n\n\n\n<p class=\"eplus-wrapper\"><strong>1. For the Pythagorean triple (p, 15, 17) , what is the value of p?<\/strong><\/p>\n\n\n\n<p class=\"eplus-wrapper\"><strong>Solution:&nbsp;<\/strong><\/p>\n\n\n\n<p class=\"eplus-wrapper\">Using the Pythagoras\u2019 theorem formula, let us substitute the values in the equation: $a^{2} + b^{2} =c^{2}$, where, $a = p, b = 15, c = 17$. (Since the hypotenuse is the longest side, we have taken $c = 17)$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$c^{2} = a^{2} +b^{2}$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$17^{2} = p^{2} +15^{2}$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$289= p^{2} + 225$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$p^{2} = 64$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$p = 8$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Therefore, the value of $p = 8$<\/p>\n\n\n\n<p class=\"eplus-wrapper\"><strong>2. Does (5, 12, 13) satisfy the Pythagorean theorem? Is it a Pythagorean triple?<\/strong><\/p>\n\n\n\n<p class=\"eplus-wrapper\"><strong>Solution:&nbsp;<\/strong><\/p>\n\n\n\n<p class=\"eplus-wrapper\">Using the Pythagoras\u2019 theorem formula, let us substitute the values in the equation: $c^{2} = a^{2} +b^{2}$ , where, $a = 5, b = 12, c = 13$. (Since hypotenuse is the longest side, we have taken $c = 13)$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$c^{2} = a^{2} +b^{2}$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$13^{2} = 5^{2} +12^{2}$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$169 = 25 + 144$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$169 = 169$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Hence, (5, 12, 13) satisfies the Pythagorean theorem.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Therefore it is a Pythagorean triple.<\/p>\n\n\n\n<p class=\"eplus-wrapper\"><strong>3. Check if (4, 5, 6) is a Pythagorean triple.<\/strong><\/p>\n\n\n\n<p class=\"eplus-wrapper\"><strong>Solution:&nbsp;<\/strong><\/p>\n\n\n\n<p class=\"eplus-wrapper\">Using the Pythagoras\u2019 theorem formula, let us substitute the values in the equation: $c^{2} = a^{2} +b^{2}$ , where $a = 4, b = 5, c = 6$. (Since hypotenuse is the longest side, we have taken $c = 6)$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$c^{2} = a^{2} +b^{2}$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$6^{2} =4^{2} +5^{2}$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$c^{2} = a^{2} +b^{2}$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$36 \\neq 41$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Hence $(4, 5, 6)$ is not a Pythagorean triple.<\/p>\n\n\n\n<p class=\"eplus-wrapper\"><strong>4. Find the two numbers in a Pythagorean triple if one number is 9.<\/strong><\/p>\n\n\n\n<p class=\"eplus-wrapper\"><strong>Solution: <\/strong>First, we need to check if the given number is odd or even.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Here, 9 is an odd number.&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Hence the other two numbers in the triple will be:<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$\\frac{x^{2}}{2} \u2212 0.5 = \\frac{81}{2} \u2212 0.5 = 40.5 \u2212 0.5 = 40$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$\\frac{x^{2}}{2} + 0.5 = \\frac{81}{2} + 0.5 = 40.5 + 0.5 = 41$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Hence the triple $(9, 40, 41)$ will be formed.<\/p>\n\n\n\n<p class=\"eplus-wrapper\"><strong>5. Is 9-12-15 a Pythagorean triple?<\/strong><\/p>\n\n\n\n<p class=\"eplus-wrapper\"><strong>Solution:&nbsp;<\/strong><\/p>\n\n\n\n<p class=\"eplus-wrapper\">$15^{2} = 225$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$12^{2} + 9^{2} = 144 + 81 = 225$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Thus, $12^{2} +9^{2} = 15^{2}$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">The given three integers satisfy the Pythagoras\u2019 theorem formula.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">So, it is a Pythagorean triple.<\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"15-practice-problems\">Practice Problems<\/h2>\n\n\n\n<p class=\"eplus-wrapper\"><div class=\"spq_wrapper\"><h2 style=\"display:none;\">Pythagorean Triples<\/h2><p style=\"display:none;\">Attend this quiz & Test your knowledge.<\/p><div class=\"spq_question_wrapper\" data-answer=\"0\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">1<\/span><h3 class=\"sqp_question_text\">There is a Pythagorean triple: (p, 12, 13) , what is the value of p.<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">5<\/div><div class=\"spq_answer_block\" data-value=\"1\">7<\/div><div class=\"spq_answer_block\" data-value=\"2\">10<\/div><div class=\"spq_answer_block\" data-value=\"3\">15<\/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: 5<br\/>Using the Pythagoras\u2019 theorem formula, let us substitute the values in the equation: $c^{2} = a^{2} +b^{2}$, where, $a = p, b = 12, c = 13$. (Since hypotenuse is the longest side, we have taken $c = 13)$<br>\r\n$c^{2} =a^{2} +b^{2}$<br>\r\n$13^{2} =p^{2} +12^{2}$<br>\r\n$169 = p^{2} + 144$<br>\r\n$p^{2} = 25$<br>\r\n$p = 5$<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"0\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">2<\/span><h3 class=\"sqp_question_text\">$(9, 12, c)$ becomes a Pythagorean triple if the hypotenuse $c =$ ?<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">15<\/div><div class=\"spq_answer_block\" data-value=\"1\">13<\/div><div class=\"spq_answer_block\" data-value=\"2\">17<\/div><div class=\"spq_answer_block\" data-value=\"3\">11<\/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: 15<br\/>Using the Pythagoras\u2019 theorem formula, let us substitute the values in the equation: $c^{2} = a^{2} +b^{2}$, where, $a = 9, b = 12$.<br>\r\nc is the hypotenuse.<br>\r\n$c^{2} = a^{2} +b^{2}$<br>\r\n$c^{2} = 9^{2} + 12^{2}$<br>\r\n$c^{2}  = 81 + 144$<br>\r\n$c^{2}  = 225$<br>\r\n$c = 15$ <br>\r\nHence, $(9, 12, 15)$ satisfies the Pythagorean theorem.<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"0\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">3<\/span><h3 class=\"sqp_question_text\">Identify a triangular number.<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">3<\/div><div class=\"spq_answer_block\" data-value=\"1\">4<\/div><div class=\"spq_answer_block\" data-value=\"2\">5<\/div><div class=\"spq_answer_block\" data-value=\"3\">7<\/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<br\/>3 is the second triangular number.<br>\r\n$1 + 2 = 3$<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"1\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">4<\/span><h3 class=\"sqp_question_text\">What are the other two numbers in a Pythagorean triple if one number is 5?<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">10, 11<\/div><div class=\"spq_answer_block\" data-value=\"1\">12, 13<\/div><div class=\"spq_answer_block\" data-value=\"2\">12, 15<\/div><div class=\"spq_answer_block\" data-value=\"3\">13, 16<\/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: 12, 13<br\/>First we need to check if the given number is odd or even.<br>\r\nHere, 5 is an odd number.<br>\r\nHence the other two numbers in the triple will be:<br>\r\n$\\frac{x^{2}}{2}$ $\u2013$ $0.5 = \\frac{25}{2}$ $\u2013$ $0.5 = 12.5$ $\u2013$ $0.5 = 12$<br>\r\n$\\frac{x^{2}}{2} + 0.5 = \\frac{81}{2} + 0.5 = 12.5 + 0.5 = 13$<br>\r\nHence the triple $(5, 12, 13)$ will be formed.<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"3\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">5<\/span><h3 class=\"sqp_question_text\">What are the other two numbers in the triple if one number is 4?<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">6, 7<\/div><div class=\"spq_answer_block\" data-value=\"1\">1, 2<\/div><div class=\"spq_answer_block\" data-value=\"2\">7, 8<\/div><div class=\"spq_answer_block\" data-value=\"3\">3, 5<\/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, 5<br\/>First, we need to check if the given number is odd or even.<br>\r\nHere, 4 is an even number.<br>\r\nHence the other two numbers in the triple will be:<br>\r\n$\\frac{x^{2}}{2}$ $-$ $1 = \\frac{4^{2}}{2}$ $\u2013$ $1=22$ $\u2013$ $1=4$ $\u2013$ $1=3$<br>\r\n$\\frac{x^{2}}{2} + 1 = \\frac{4^{2}}{2} + 1 = 22 + 1 = 4 + 1 = 5$.<br>\r\nHence the triple $(3, 4, 5)$ will be formed.<\/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\": \"Pythagorean Triples\",        \n        \"about\": {\n                \"@type\": \"Thing\",\n                \"name\": \"Pythagorean Triples\"\n        },  \n        \"hasPart\": [{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"There is a Pythagorean triple: (p, 12, 13) , what is the value of p.\",\n                    \"text\": \"There is a Pythagorean triple: (p, 12, 13) , what is the value of p.\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"Using the Pythagoras\u2019 theorem formula, let us substitute the values in the equation: $$c^{2} = a^{2} +b^{2}$$, where, $$a = p, b = 12, c = 13$$. (Since hypotenuse is the longest side, we have taken $$c = 13)$$<br>\r\n$$c^{2} =a^{2} +b^{2}$$<br>\r\n$$13^{2} =p^{2} +12^{2}$$<br>\r\n$$169 = p^{2} + 144$$<br>\r\n$$p^{2} = 25$$<br>\r\n$$p = 5$$\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 1,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"7\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"Using the Pythagoras\u2019 theorem formula, let us substitute the values in the equation: $$c^{2} = a^{2} +b^{2}$$, where, $$a = p, b = 12, c = 13$$. (Since hypotenuse is the longest side, we have taken $$c = 13)$$<br>\r\n$$c^{2} =a^{2} +b^{2}$$<br>\r\n$$13^{2} =p^{2} +12^{2}$$<br>\r\n$$169 = p^{2} + 144$$<br>\r\n$$p^{2} = 25$$<br>\r\n$$p = 5$$\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 2,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"10\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"Using the Pythagoras\u2019 theorem formula, let us substitute the values in the equation: $$c^{2} = a^{2} +b^{2}$$, where, $$a = p, b = 12, c = 13$$. (Since hypotenuse is the longest side, we have taken $$c = 13)$$<br>\r\n$$c^{2} =a^{2} +b^{2}$$<br>\r\n$$13^{2} =p^{2} +12^{2}$$<br>\r\n$$169 = p^{2} + 144$$<br>\r\n$$p^{2} = 25$$<br>\r\n$$p = 5$$\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 3,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"15\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"Using the Pythagoras\u2019 theorem formula, let us substitute the values in the equation: $$c^{2} = a^{2} +b^{2}$$, where, $$a = p, b = 12, c = 13$$. (Since hypotenuse is the longest side, we have taken $$c = 13)$$<br>\r\n$$c^{2} =a^{2} +b^{2}$$<br>\r\n$$13^{2} =p^{2} +12^{2}$$<br>\r\n$$169 = p^{2} + 144$$<br>\r\n$$p^{2} = 25$$<br>\r\n$$p = 5$$\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 0,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"5\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"Using the Pythagoras\u2019 theorem formula, let us substitute the values in the equation: $$c^{2} = a^{2} +b^{2}$$, where, $$a = p, b = 12, c = 13$$. (Since hypotenuse is the longest side, we have taken $$c = 13)$$<br>\r\n$$c^{2} =a^{2} +b^{2}$$<br>\r\n$$13^{2} =p^{2} +12^{2}$$<br>\r\n$$169 = p^{2} + 144$$<br>\r\n$$p^{2} = 25$$<br>\r\n$$p = 5$$\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"Using the Pythagoras\u2019 theorem formula, let us substitute the values in the equation: $$c^{2} = a^{2} +b^{2}$$, where, $$a = p, b = 12, c = 13$$. (Since hypotenuse is the longest side, we have taken $$c = 13)$$<br>\r\n$$c^{2} =a^{2} +b^{2}$$<br>\r\n$$13^{2} =p^{2} +12^{2}$$<br>\r\n$$169 = p^{2} + 144$$<br>\r\n$$p^{2} = 25$$<br>\r\n$$p = 5$$\"\n                      }\n                    } \n\n                    },{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"$$(9, 12, c)$$ becomes a Pythagorean triple if the hypotenuse $$c =$$ ?\",\n                    \"text\": \"$$(9, 12, c)$$ becomes a Pythagorean triple if the hypotenuse $$c =$$ ?\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"Using the Pythagoras\u2019 theorem formula, let us substitute the values in the equation: $$c^{2} = a^{2} +b^{2}$$, where, $$a = 9, b = 12$$.<br>\r\nc is the hypotenuse.<br>\r\n$$c^{2} = a^{2} +b^{2}$$<br>\r\n$$c^{2} = 9^{2} + 12^{2}$$<br>\r\n$$c^{2}  = 81 + 144$$<br>\r\n$$c^{2}  = 225$$<br>\r\n$$c = 15$$ <br>\r\nHence, $$(9, 12, 15)$$ satisfies the Pythagorean theorem.\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 1,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"13\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"Using the Pythagoras\u2019 theorem formula, let us substitute the values in the equation: $$c^{2} = a^{2} +b^{2}$$, where, $$a = 9, b = 12$$.<br>\r\nc is the hypotenuse.<br>\r\n$$c^{2} = a^{2} +b^{2}$$<br>\r\n$$c^{2} = 9^{2} + 12^{2}$$<br>\r\n$$c^{2}  = 81 + 144$$<br>\r\n$$c^{2}  = 225$$<br>\r\n$$c = 15$$ <br>\r\nHence, $$(9, 12, 15)$$ satisfies the Pythagorean theorem.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 2,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"17\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"Using the Pythagoras\u2019 theorem formula, let us substitute the values in the equation: $$c^{2} = a^{2} +b^{2}$$, where, $$a = 9, b = 12$$.<br>\r\nc is the hypotenuse.<br>\r\n$$c^{2} = a^{2} +b^{2}$$<br>\r\n$$c^{2} = 9^{2} + 12^{2}$$<br>\r\n$$c^{2}  = 81 + 144$$<br>\r\n$$c^{2}  = 225$$<br>\r\n$$c = 15$$ <br>\r\nHence, $$(9, 12, 15)$$ satisfies the Pythagorean theorem.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 3,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"11\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"Using the Pythagoras\u2019 theorem formula, let us substitute the values in the equation: $$c^{2} = a^{2} +b^{2}$$, where, $$a = 9, b = 12$$.<br>\r\nc is the hypotenuse.<br>\r\n$$c^{2} = a^{2} +b^{2}$$<br>\r\n$$c^{2} = 9^{2} + 12^{2}$$<br>\r\n$$c^{2}  = 81 + 144$$<br>\r\n$$c^{2}  = 225$$<br>\r\n$$c = 15$$ <br>\r\nHence, $$(9, 12, 15)$$ satisfies the Pythagorean theorem.\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 0,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"15\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"Using the Pythagoras\u2019 theorem formula, let us substitute the values in the equation: $$c^{2} = a^{2} +b^{2}$$, where, $$a = 9, b = 12$$.<br>\r\nc is the hypotenuse.<br>\r\n$$c^{2} = a^{2} +b^{2}$$<br>\r\n$$c^{2} = 9^{2} + 12^{2}$$<br>\r\n$$c^{2}  = 81 + 144$$<br>\r\n$$c^{2}  = 225$$<br>\r\n$$c = 15$$ <br>\r\nHence, $$(9, 12, 15)$$ satisfies the Pythagorean theorem.\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"Using the Pythagoras\u2019 theorem formula, let us substitute the values in the equation: $$c^{2} = a^{2} +b^{2}$$, where, $$a = 9, b = 12$$.<br>\r\nc is the hypotenuse.<br>\r\n$$c^{2} = a^{2} +b^{2}$$<br>\r\n$$c^{2} = 9^{2} + 12^{2}$$<br>\r\n$$c^{2}  = 81 + 144$$<br>\r\n$$c^{2}  = 225$$<br>\r\n$$c = 15$$ <br>\r\nHence, $$(9, 12, 15)$$ satisfies the Pythagorean theorem.\"\n                      }\n                    } \n\n                    },{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"Identify a triangular number.\",\n                    \"text\": \"Identify a triangular number.\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"3 is the second triangular number.<br>\r\n$$1 + 2 = 3$$\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 1,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"4\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"3 is the second triangular number.<br>\r\n$$1 + 2 = 3$$\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 2,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"5\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"3 is the second triangular number.<br>\r\n$$1 + 2 = 3$$\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 3,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"7\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"3 is the second triangular number.<br>\r\n$$1 + 2 = 3$$\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 0,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"3\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"3 is the second triangular number.<br>\r\n$$1 + 2 = 3$$\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"3 is the second triangular number.<br>\r\n$$1 + 2 = 3$$\"\n                      }\n                    } \n\n                    },{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"What are the other two numbers in a Pythagorean triple if one number is 5?\",\n                    \"text\": \"What are the other two numbers in a Pythagorean triple if one number is 5?\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"First we need to check if the given number is odd or even.<br>\r\nHere, 5 is an odd number.<br>\r\nHence the other two numbers in the triple will be:<br>\r\n$$\\\\frac{x^{2}}{2}$$ $$\u2013$$ $$0.5 = \\\\frac{25}{2}$$ $$\u2013$$ $$0.5 = 12.5$$ $$\u2013$$ $$0.5 = 12$$<br>\r\n$$\\\\frac{x^{2}}{2} + 0.5 = \\\\frac{81}{2} + 0.5 = 12.5 + 0.5 = 13$$<br>\r\nHence the triple $$(5, 12, 13)$$ will be formed.\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 0,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"10, 11\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"First we need to check if the given number is odd or even.<br>\r\nHere, 5 is an odd number.<br>\r\nHence the other two numbers in the triple will be:<br>\r\n$$\\\\frac{x^{2}}{2}$$ $$\u2013$$ $$0.5 = \\\\frac{25}{2}$$ $$\u2013$$ $$0.5 = 12.5$$ $$\u2013$$ $$0.5 = 12$$<br>\r\n$$\\\\frac{x^{2}}{2} + 0.5 = \\\\frac{81}{2} + 0.5 = 12.5 + 0.5 = 13$$<br>\r\nHence the triple $$(5, 12, 13)$$ will be formed.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 2,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"12, 15\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"First we need to check if the given number is odd or even.<br>\r\nHere, 5 is an odd number.<br>\r\nHence the other two numbers in the triple will be:<br>\r\n$$\\\\frac{x^{2}}{2}$$ $$\u2013$$ $$0.5 = \\\\frac{25}{2}$$ $$\u2013$$ $$0.5 = 12.5$$ $$\u2013$$ $$0.5 = 12$$<br>\r\n$$\\\\frac{x^{2}}{2} + 0.5 = \\\\frac{81}{2} + 0.5 = 12.5 + 0.5 = 13$$<br>\r\nHence the triple $$(5, 12, 13)$$ will be formed.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 3,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"13, 16\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"First we need to check if the given number is odd or even.<br>\r\nHere, 5 is an odd number.<br>\r\nHence the other two numbers in the triple will be:<br>\r\n$$\\\\frac{x^{2}}{2}$$ $$\u2013$$ $$0.5 = \\\\frac{25}{2}$$ $$\u2013$$ $$0.5 = 12.5$$ $$\u2013$$ $$0.5 = 12$$<br>\r\n$$\\\\frac{x^{2}}{2} + 0.5 = \\\\frac{81}{2} + 0.5 = 12.5 + 0.5 = 13$$<br>\r\nHence the triple $$(5, 12, 13)$$ will be formed.\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 1,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"12, 13\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"First we need to check if the given number is odd or even.<br>\r\nHere, 5 is an odd number.<br>\r\nHence the other two numbers in the triple will be:<br>\r\n$$\\\\frac{x^{2}}{2}$$ $$\u2013$$ $$0.5 = \\\\frac{25}{2}$$ $$\u2013$$ $$0.5 = 12.5$$ $$\u2013$$ $$0.5 = 12$$<br>\r\n$$\\\\frac{x^{2}}{2} + 0.5 = \\\\frac{81}{2} + 0.5 = 12.5 + 0.5 = 13$$<br>\r\nHence the triple $$(5, 12, 13)$$ will be formed.\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"First we need to check if the given number is odd or even.<br>\r\nHere, 5 is an odd number.<br>\r\nHence the other two numbers in the triple will be:<br>\r\n$$\\\\frac{x^{2}}{2}$$ $$\u2013$$ $$0.5 = \\\\frac{25}{2}$$ $$\u2013$$ $$0.5 = 12.5$$ $$\u2013$$ $$0.5 = 12$$<br>\r\n$$\\\\frac{x^{2}}{2} + 0.5 = \\\\frac{81}{2} + 0.5 = 12.5 + 0.5 = 13$$<br>\r\nHence the triple $$(5, 12, 13)$$ will be formed.\"\n                      }\n                    } \n\n                    },{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"What are the other two numbers in the triple if one number is 4?\",\n                    \"text\": \"What are the other two numbers in the triple if one number is 4?\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"First, we need to check if the given number is odd or even.<br>\r\nHere, 4 is an even number.<br>\r\nHence the other two numbers in the triple will be:<br>\r\n$$\\\\frac{x^{2}}{2}$$ $$-$$ $$1 = \\\\frac{4^{2}}{2}$$ $$\u2013$$ $$1=22$$ $$\u2013$$ $$1=4$$ $$\u2013$$ $$1=3$$<br>\r\n$$\\\\frac{x^{2}}{2} + 1 = \\\\frac{4^{2}}{2} + 1 = 22 + 1 = 4 + 1 = 5$$.<br>\r\nHence the triple $$(3, 4, 5)$$ will be formed.\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 0,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"6, 7\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"First, we need to check if the given number is odd or even.<br>\r\nHere, 4 is an even number.<br>\r\nHence the other two numbers in the triple will be:<br>\r\n$$\\\\frac{x^{2}}{2}$$ $$-$$ $$1 = \\\\frac{4^{2}}{2}$$ $$\u2013$$ $$1=22$$ $$\u2013$$ $$1=4$$ $$\u2013$$ $$1=3$$<br>\r\n$$\\\\frac{x^{2}}{2} + 1 = \\\\frac{4^{2}}{2} + 1 = 22 + 1 = 4 + 1 = 5$$.<br>\r\nHence the triple $$(3, 4, 5)$$ will be formed.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 1,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"1, 2\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"First, we need to check if the given number is odd or even.<br>\r\nHere, 4 is an even number.<br>\r\nHence the other two numbers in the triple will be:<br>\r\n$$\\\\frac{x^{2}}{2}$$ $$-$$ $$1 = \\\\frac{4^{2}}{2}$$ $$\u2013$$ $$1=22$$ $$\u2013$$ $$1=4$$ $$\u2013$$ $$1=3$$<br>\r\n$$\\\\frac{x^{2}}{2} + 1 = \\\\frac{4^{2}}{2} + 1 = 22 + 1 = 4 + 1 = 5$$.<br>\r\nHence the triple $$(3, 4, 5)$$ will be formed.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 2,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"7, 8\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"First, we need to check if the given number is odd or even.<br>\r\nHere, 4 is an even number.<br>\r\nHence the other two numbers in the triple will be:<br>\r\n$$\\\\frac{x^{2}}{2}$$ $$-$$ $$1 = \\\\frac{4^{2}}{2}$$ $$\u2013$$ $$1=22$$ $$\u2013$$ $$1=4$$ $$\u2013$$ $$1=3$$<br>\r\n$$\\\\frac{x^{2}}{2} + 1 = \\\\frac{4^{2}}{2} + 1 = 22 + 1 = 4 + 1 = 5$$.<br>\r\nHence the triple $$(3, 4, 5)$$ will be formed.\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 3,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"3, 5\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"First, we need to check if the given number is odd or even.<br>\r\nHere, 4 is an even number.<br>\r\nHence the other two numbers in the triple will be:<br>\r\n$$\\\\frac{x^{2}}{2}$$ $$-$$ $$1 = \\\\frac{4^{2}}{2}$$ $$\u2013$$ $$1=22$$ $$\u2013$$ $$1=4$$ $$\u2013$$ $$1=3$$<br>\r\n$$\\\\frac{x^{2}}{2} + 1 = \\\\frac{4^{2}}{2} + 1 = 22 + 1 = 4 + 1 = 5$$.<br>\r\nHence the triple $$(3, 4, 5)$$ will be formed.\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"First, we need to check if the given number is odd or even.<br>\r\nHere, 4 is an even number.<br>\r\nHence the other two numbers in the triple will be:<br>\r\n$$\\\\frac{x^{2}}{2}$$ $$-$$ $$1 = \\\\frac{4^{2}}{2}$$ $$\u2013$$ $$1=22$$ $$\u2013$$ $$1=4$$ $$\u2013$$ $$1=3$$<br>\r\n$$\\\\frac{x^{2}}{2} + 1 = \\\\frac{4^{2}}{2} + 1 = 22 + 1 = 4 + 1 = 5$$.<br>\r\nHence the triple $$(3, 4, 5)$$ will be formed.\"\n                      }\n                    } \n\n                    }]}<\/script><\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"16-frequently-asked-questions\">Frequently Asked Questions<\/h2>\n\n\n<div class=\"wp-block-ub-content-toggle\" id=\"ub-content-toggle-b6bf3f08-d343-46b8-9cfb-654e8c941232\" 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-b6bf3f08-d343-46b8-9cfb-654e8c941232\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-b6bf3f08-d343-46b8-9cfb-654e8c941232\"><strong>What is a triple?<\/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-b6bf3f08-d343-46b8-9cfb-654e8c941232\">\n\n<p class=\"eplus-wrapper\">A set of three numbers is called a triple.<\/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-b6bf3f08-d343-46b8-9cfb-654e8c941232\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-b6bf3f08-d343-46b8-9cfb-654e8c941232\"><strong>Can Pythagorean triples include numbers with decimals?<\/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-b6bf3f08-d343-46b8-9cfb-654e8c941232\">\n\n<p class=\"eplus-wrapper\">No, only positive integer numbers can satisfy Pythagorean triples.<\/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-b6bf3f08-d343-46b8-9cfb-654e8c941232\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-b6bf3f08-d343-46b8-9cfb-654e8c941232\"><strong>When was the Pythagorean theorem invented?<\/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-b6bf3f08-d343-46b8-9cfb-654e8c941232\">\n\n<p class=\"eplus-wrapper\">The Pythagorean theorem was invented around 1900 BCE.<\/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-b6bf3f08-d343-46b8-9cfb-654e8c941232\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-b6bf3f08-d343-46b8-9cfb-654e8c941232\"><strong>Who was Pythagoras?<\/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-b6bf3f08-d343-46b8-9cfb-654e8c941232\">\n\n<p class=\"eplus-wrapper\">Pythagoras was a Greek philosopher and mathematician who made important developments in mathematics and astronomy.<\/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-4-b6bf3f08-d343-46b8-9cfb-654e8c941232\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-b6bf3f08-d343-46b8-9cfb-654e8c941232\"><strong>How many proofs of the Pythagorean theorem are there?<\/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-4-b6bf3f08-d343-46b8-9cfb-654e8c941232\">\n\n<p class=\"eplus-wrapper\">There are about 371 Pythagorean theorem proofs.<\/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-5-b6bf3f08-d343-46b8-9cfb-654e8c941232\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-b6bf3f08-d343-46b8-9cfb-654e8c941232\"><strong>How many Pythagorean triples are there?<\/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-5-b6bf3f08-d343-46b8-9cfb-654e8c941232\">\n\n<p class=\"eplus-wrapper\">There are an infinite number of Pythagorean triples.<\/p>\n\n<\/div><\/div>\n<\/div>\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"17-related-articles\">Related Articles<\/h2>\n\n\n\n<ul class=\"eplus-wrapper wp-block-list\">\n<li class=\"eplus-wrapper\"><a href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/geometry\/right-triangle\">Right Angled Triangle<\/a><\/li>\n\n\n\n<li class=\"eplus-wrapper\"><a href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/geometry\/acute-angle\">Acute Angle<\/a><\/li>\n\n\n\n<li class=\"eplus-wrapper\"><a href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/algebra\/formula\">Formula<\/a><\/li>\n\n\n\n<li class=\"eplus-wrapper\"><a href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/geometry\/geometry\">Geometry<\/a><\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Introduction to Pythagorean Triples Pythagorean triples are three positive integers which satisfy the Pythagoras\u2019 theorem. Pythagoras\u2019 theorem states that in any right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides of the right triangle. Any three positive integers that fully satisfy the Pythagorean &#8230; <a title=\"Pythagorean Triples\" class=\"read-more\" href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/pythagorean-triples\" aria-label=\"More on Pythagorean Triples\">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-23497","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\/23497","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=23497"}],"version-history":[{"count":11,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts\/23497\/revisions"}],"predecessor-version":[{"id":37400,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts\/23497\/revisions\/37400"}],"wp:attachment":[{"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/media?parent=23497"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/categories?post=23497"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/tags?post=23497"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}