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# Special Right Triangles – Explanation & Examples

Now you know a **triangle is a two-dimensional polygon** with **3 sides**, **3 angles,** and **3 vertices**. In this article, we are going to learn other types of triangles known as special right triangles. Before we can start, let‘s recall about a right triangle.

**What is a Right triangle?**

The term “**right**” refers to the Latin word “**rectus,**” meaning *upright.* Therefore, a right triangle is a triangle whose one angle is 90 degrees (**right angle**). Right triangles are indicated with a box at the location of the right angle.

The longest side of the right triangle on the opposite side of the right angle is known as the hypotenuse. The other two sides of the triangle are known as legs. The horizontal leg is the base, and the vertical leg is the height of a right triangle.

Illustration:

## What is a Special Right Triangle?

Special right triangles are triangles whose sides are in a particular ratio, known as Pythagorean Triples. In geometry, the **Pythagorean Theorem** is a statement that shows the relationship of the sides of a right triangle.

The equation of a right triangle is given by **a ^{2} + b^{2} = c^{2}**, where either a or b is the height and base of the triangle and c is the hypotenuse. Using the Pythagorean Theorem, finding the missing side of a triangle is pretty simple and easy.

*The two special right triangles include:*

- 45°; 45°; 90° Triangle
- 30°; 60°; 90° Triangle

Let’s have a brief overview of these special right triangles as we will see them in detail in the next articles.

**The 45°; 45°; 90° Triangle**

This is a **special right triangle** whose angles are 45°, 45°, and 90°. The base to height ratio to the hypotenuse of this triangle is 1: 1: √2.

Base: Height: Hypotenuse = x: x: x√2 = 1: 1: √2.

In other words, a 45°; 45°; 90° triangle can also be isosceles. An isosceles triangle is a triangle in which two the lengths of its two sides are equal and also the two of its angles are equal.

By using the equation of a right triangle a^{2} + b^{2} = c^{2}, we can calculate the hypotenuse of, a 45°; 45°; 90° triangle as follows:

Since, a 45°; 45°; 90° triangle is also an isosceles triangle;

let a = b = x;

x^{2} + x^{2} = 2x^{2}

Find the square root of each term in the equation

√x^{2 }+ √x^{2 }= √(2x^{2})

x + x = x √2

Therefore, the hypotenuse of a 45°; 45°; 90° triangle is x √2

**The 30°; 60°; 90° Triangle**

This is a special type of right triangle whose angles are 30°; 60°; 90°. The ratio of the lengths of the sides is x: x√3: 2x.

## How to Solve Special Right Triangles?

Solving special right triangles means finding the missing lengths of the sides. Instead of using the Pythagorean Theorem, we can use the special right triangle ratios to perform calculations.

*Let’s work out a couple of examples.*

*Example 1*

The longer side of a 30°; 60°; 90° right triangle is given by 8√3 cm. What is the measure of its height and hypotenuse?

__Solution__

The best way to solve these kinds of problems is to sketch the triangles:

The ratio of a 30°; 60°; 90° right triangle is x: x√3: 2x. In this case, x and x√3 are the shorter and longer sides, respectively, while 2x is the hypotenuse.

Therefore, x√3 = 8√3 cm

Square both sides of the equation.

⇒ (x√3)^{2} = (8√3)^{2}

⇒ 3x^{2} = 64 * 3

⇒ x ^{2} = 64

Find the square of both sides.

√x^{2} = √64

x = 8cm

Substitute.

2x = 2 * 8 = 16cm.

Therefore, the shorter side is 8cm, and the hypotenuse is 16cm.

*Example 2*

The hypotenuse of a 45°; 45°; 90° triangle is 6√2 mm. Calculate the length of its base and height.

__Solution__

Ratio of a 45°; 45°; 90° triangle is x: x: x√2. So, we have;

⇒x√2 = 6√2 mm

Square both sides of the equation.

⇒(x√2)^{2} = (6√2)^{2} mm

⇒ 2x^{2} = 36 * 2

⇒ 2x^{2 }= 72

x^{2} = 36

Find the square root.

x = 6mm

Substitute x = 6mm in the ratio.

Hence, the base and height of the right triangle are 6mm each.

*Example 3*

If the diagonal of a right triangle is 8 cm, find the other two sides of the triangle’s lengths given that one of its angles is 30 degrees.

__Solution__

This is a 30°-60°-90° triangle. Therefore, we use the ratio of x: x√3:2x.

Given, the diagonal = hypotenuse = 8cm.

⇒2x = 8 cm

⇒ x = 4cm

Substitute.

x√3 = 4√3 cm

The shorter side of the right triangle is 4cm, and the longer side is 4√3 cm.

*Example 4*

Find the hypotenuse of a 30°- 60°- 90° triangle whose longer side is 6 inches.

__Solution__

Ratio = x: x√3:2x.

⇒ x√3 = 6 inches.

Square both sides

⇒ (x√3)^{2} = 36

⇒ 3x^{2} = 36

x^{2} = 12

x = 2√3 inches.

*Example 5*

A ladder leaning against a wall makes an angle of 30 degrees with the ground. If the length of the ladder is 9 m, find;

- The height of the wall.
- Calculate the length between the foot of the ladder and the wall.

__Solution__

Given that one angle is 30 degrees, this must be a 60°- 60°- 90°right triangle.

Ratio = x: x√3:2x.

⇒ 2x = 9

⇒ x = 9/2

= 4.5

Substitute.

- The height of the wall = 4.5m
- x√3 = 4.5√3m

*Practice Questions*

- If the length of one side of an equilateral triangle is 15 m, what is the length of that triangle’s altitude?
- If the length of the diagonal of the square is 10 units, what is the square area?
- If an altitude of an equilateral triangle is 22 cm, what is the length of a side of an equilateral triangle?