## What Is a Hypotenuse?

**A right-angled triangle is a triangle that has one interior angle, which measures 90 degrees. The side opposite to the right angle in a right-angled triangle is known as the hypotenuse. It is the longest side.**

### Definition of Hypotenuse?

The sides of a right triangle are base, perpendicular, and hypotenuse. As mentioned earlier, the hypotenuse of a right triangle lies opposite to the right angle. In a right-angled triangle, the sides other than the hypotenuse which determine the right angle are also referred to as “legs.” The side that makes a right angle with the base is called the perpendicular.

Thus, the definition of the hypotenuse in geometry can be given as the longest side in a right triangle that lies opposite to the right angle.

## Theorem for the Hypotenuse

The famous Pythagoras’ theorem defines the **hypotenuse** theorem. As per this theorem, in a right-angled triangle, the square of the hypotenuse side is equal to the sum of the squares of the other two sides of the triangle, i.e., the base and perpendicular side.

**Base² **$+$** Perpendicular² **$=$** Hypotenuse ^{2}**

## Proof of the Hypotenuse Theorem

Do you wonder how the **hypotenuse** theorem was derived? Let’s understand its proof.

The triangle ABC is a right-angled triangle such that m$\angle$B $= 90^{\circ}$.

AB is the base side, BC is the perpendicular side, and AC is the hypotenuse side.

We need to prove that AB² $+$ BC² $=$ AC²

For this, we draw a line from B to meet the side AC at point D.

Using the similar triangle theorem, we get,

$\Delta$ADB ~ $\Delta$ABC by AAA similarity

Thus, its corresponding sides are proportional.

$\frac{AD}{AB} = \frac{AB}{AC}$

AB² $=$ AD $\times$ AC – (i)

Now, using the similar triangle theorem again, we get,

$\Delta$BDC ~ $\Delta$ABC

So, $\frac{AD}{AB} = \frac{AB}{AC}$

That is BC² $=$ CD $\times$AC – (ii)

Now, by adding (i) and (ii), we will get,

AB² $+$ BC² $=$ (AD $\times$ AC) + (CD $\times$ AC)

AB² $+$ BC² $=$ AC (AD $+$ CD)

AB² $+$ BC² $=$ AC (AC)

AB² $+$ BC² $=$ AC²

Or

AC² $=$ AB² $+$ BC²

We can also write this as

**Hypotenuse² **$=$** Base² **$+$** Perpendicular²**

## Formula of the Hypotenuse

How to find the hypotenuse of a right triangle? We can use the formula that includes the length of the base and the perpendicular.

The **formula for a hypotenuse** is

**Hypotenuse **$= \sqrt{Base² + Perpendicular²}$

In a right-angled triangle ABC,

Hypotenuse $= c$

Base $= b$

Perpendicular or Altitude $= a$

** **$a^{2} + b^{2} = c^{2}$

**How to Calculate the Length of the Hypotenuse of a Right Triangle?**

Let’s understand the steps for the **calculation of the length of the hypotenuse of a right triangle** when base and height are given.

Example:

We know that the base side, i.e., AB $= 4$ inches, and the perpendicular side, i.e., BC $= 3$ inches.

To find the length of the **hypotenuse, **we need to use the above formula.

**Hypotenuse **$=$** Base² **$+$** Perpendicular²**

AC $= \sqrt{(AB)² + (BC)²}$

AC $= \sqrt{(4)² + (3)²}$

AC $= \sqrt{16 + 9}$

AC $= \sqrt{25}$

AC $= 5$ inches.

Thus, the **hypotenuse** of the right triangle ABC is 5 inches.

## How To Find the Altitude Drawn on a Hypotenuse?

The altitude on a **hypotenuse** is the perpendicular drawn on the hypotenuse that connects a right triangle’s hypotenuse to its opposite vertex.

To find the altitude on a hypotenuse, let’s take an example.

From the right-angled triangle ABC, we know that,

Base $= 6$ inches.

Perpendicular $= 8$ inches.

Hypotenuse $= 10$ inches.

To find the altitude, we divide the shortest side by the hypotenuse of $\Delta$ABC, i.e., Base/Hypotenuse $= \frac{6}{10} = \frac{3}{5}$.

Finally, we multiply this value by the value of the remaining side of the $\Delta$ABC, i.e.,

$\frac{3}{5} \times $ perpendicular $= \frac{3}{5} \times 8 = \frac{24}{5} = 4.8$.

Thus, the value of the altitude in $\Delta$ABC is 4.8 inches.

## Area of an Isosceles Right Triangle

Generally, the area of an isosceles right triangle is calculated with the help of this simple formula:

**Area **$= \frac{1}{2}$** Base **$\times$** Perpendicular (or altitude)**

Since an isosceles right triangle has two equal sides, we can also write the formula as:

**Area **$= \frac{1}{2}\times$** (Base)²**

Or

**Area **$= \frac{1}{2}\times$** (Perpendicular)².**

For example: Calculate the area of the isosceles right triangle given below.

We know that the base side, i.e., DE $= 9$ inches.

So, the area of the isosceles right triangle $= \frac{1}{2}\times$ (Base)²

$=\frac{1}{2}\times$ (DE)²

$= \frac{1}{2}\times (9)²$

$= \frac{1}{2}\times 81$

$= 40.5$ inches².

In scenarios where only the **hypotenuse** of an isosceles right triangle is given, we can still find its area. Let’s understand this better with an example.

Example: Find the area of the isosceles right triangle whose hypotenuse is 12 feet.

We know that hypotenuse $= \sqrt{(base² + perpendicular²)}$

Since the triangle is isosceles, hypotenuse $= \sqrt{(base² + base²)}$

$12 = \sqrt{2(base)^{2}}$

$(12)² = 2(base²)$

Here, the squares on either side will cancel each other out, and we will get:

$12 = 2(base)$

$\frac{12}{2} \times = base$

$6 = base$.

Thus, the base $= 6$ feet $=$ perpendicular.

Now, since we know the sides of an isosceles triangle other than its hypotenuse, we can find its area by using the formula:

$\frac{1}{2} \times (Base)² = \frac{1}{2} \times (6)² = \frac{1}{2} \times 36 = 18$ inches².

## Fun Facts

- Did you know that the word
**hypotenuse**is derived from the Greek word*hypoteinousa*? “Hypoteinousa” means “stretching under (a right angle).”

- The relationship between altitude on a hypotenuse and a right-angled triangle is explained by the Right Triangle Altitude Theorem. As per this theorem, altitude on the hypotenuse of a right-angled triangle divides it into two congruent right-angled triangles. The two triangles are similar to the main right-angled triangle.

- In a right-angled triangle, the altitude on the hypotenuse can also be calculated with the help of basic trigonometric ratios.

## Conclusion

In geometry, the **hypotenuse** is defined as the side opposite the right angle in a right-angled triangle. Its theorem is defined by Pythagoras’ theorem (Hypotenuse² $=$ Base² $+$ Perpendicular²). This formula helps us find the hypotenuse and area of a right triangle or an isosceles right triangle in math and for real-life objects as well.

## Solved Examples on Hypotenuse

**Find the length of the hypotenuse DF.**

**Solution:**

Here, the base side (EF) $= 5$ inches and the perpendicular (DE) $= 12$ inches.

Hypotenuse $= \sqrt{(base² + perpendicular²)}$

$= \sqrt{\left\{(5)² + (12)²\right\}}$

$= \sqrt{25+144}$

$= \sqrt{169}$

$= 13$ inches.

Thus, the length of the hypotenuse (DF) is 13 inches.

**2. Hypotenuse **$= 100$** feet, Base **$= 8$** feet. Find the height of the right triangle.**

**Solution: **

We know that

Hypotenuse² $=$ Base² $+ $Perpendicular².

$100 = 8^{2} +$ Perpendicular²

Perpendicular² $= 100 \;-\;64$

Perpendicular² $= 36$

Perpendicular $= 6$ feet

**3. Find the area of a right triangle whose base and perpendicular are 10 feet each.**

**Solution: **

When the triangle is isosceles (base side = perpendicular side), the formula finding the area

$= \frac{1}{2} \times$(perpendicular)²

$= \frac{1}{2} \times (10)² $

$= \frac{1}{2} \times 100$

$= 50$ feet².

**4. Find the base of the right-angled sandwich.**

**Solution: **We know that hypotenuse $= 17$ inches and perpendicular $= 15$ inches.

Since hypotenuse² $=$ (base² $+$ perpendicular²).

So, base² $=$ (hypotenuse² $-$ perpendicular²)

base² $= (17)² \;-\; (15)²$

base² $= 289 \;-\; 225$

base² $= 64$

base² $= (8)²$

Squares on each side will cancel each other, and we will get,

Base $= 8$ inches.

**5. Determine the altitude of a right triangle whose hypotenuse, base, and perpendicular are 15, 9, and 12 inches.**

**Solution: **To find the altitude, we need to divide the shortest side (here, base) by the hypotenuse.

Base/Hypotenuse $= \frac{9}{15} = \frac{3}{5}$.

Now, we need to multiply the above value by the remaining side, i.e., the perpendicular side.

$\frac{3}{5} \times$ perpendicular $= \frac{3}{5} \times 12 = \frac{36}{5} = 7.2$

Thus, the altitude of the right triangle is 7.2 inches.

## Practice Problems on Hypotenuse

## Hypotenuse in Right Triangle

### Hypotenuse is ______________.

The hypotenuse is the side that lies opposite to the right angle.

### Find the length of the perpendicular of a right triangle whose base is 7 inches and whose hypotenuse is 25 inches.

Hypotenuse² $=$ Base² $+$ Perpendicular²

So, Perpendicular² $=$ Hypotenuse² $-$ Base²

Perpendicular² $= (25)² \;-\; (7)² = 625 \;-\; 49 = 576$

Perpendicular² $= (24)²$

Perpendicular $= 24$ inches.

### The longest side of the right-angled triangle is called ____________.

Hypotenuse is the longest side of the right-angled triangle.

### In the following image, a and b are ____.

In right triangles, the base and the perpendicular are also called legs.

### Find AC.

AC $= \sqrt{(AB)² + (BC)²}$

AC $= \sqrt{(24)² + (10)²}$

AC $= \sqrt{(576 + 100)} = \sqrt{676} = 26$ feet.

## Frequently Asked Questions on Hypotenuse

**What is the similar triangle theorem?**

The similar triangle theorem states that two triangles are similar when their corresponding angles are congruent, and their sides are proportionately equal.

**Can you find the hypotenuse of other triangles other than the right triangle?**

No. In geometry, the hypotenuse is only defined for right-angled triangles and not for any other type of triangle.

**What are the sides of a right triangle collectively called?**

The sides of a right triangle are collectively known as Pythagorean triplets.