# 45-45-90 Triangle

## What is a 45-45-90 Triangle

A 45-45-90 triangle is a special right triangle that has two 45-degree angles and one 90-degree angle.

It is also sometimes called an isosceles right triangle since it has two equal sides and two equal angles. It is a special triangle because its side lengths are always in the ratio of 1:1: √2 (x: x: x√2 for side1: side2: hypotenuse).

The 45-45-90 triangle is half of a square. This is because a square has all four angles measuring 90° each. When a square is cut diagonally, one angle remains 90° and the other two 90° angles bisected and become 45° each. The diagonal of a square becomes hypotenuse of a right triangle and the other two sides become the base and the height of the 45-45-90 triangle.

Let side 1 and side 2 of an isosceles-right be x.

We can calculate the hypotenuse of a 45-45-90 right triangle applying the Pythagoras formula a2 + b2 = c2, where a = side 1, b = side 2, and c = hypotenuse.

Hypotenuse = x2 + x2 = 2x2

x + x = x√2

## Properties and Rules

1. Has two equal side lengths and two equal angles and thus the only possible right-triangle, which is isosceles
2. The angles are in the ratio 1: 1: 2
3. The sides are in the ratio 1: 1: √2 (x: x: x√2)
4. Has one line of symmetry
5. Has no rotational symmetry

Thus, the most important rule of a 45-45-90 triangle is that it has one right angle and two other angles equal to 45°. This means, it has two sides of equal length and an unequal side called the base.

## How to Solve a 45-45-90 Triangle

Given the length of one side of a 45-45-90 triangle, we can easily calculate the other missing side lengths and also the area and perimeter of the triangle without using the Pythagorean Theorem or trigonometric functions.

Solving a 45-45-90 triangle can have two possibilities:

• Possibility 1: To calculate hypotenuse when the length of one side is given, multiply the given length by √2
• Possibility 2: To calculate the side lengths given the length of the hypotenuse, divide the hypotenuse by √2

Thus, all problems regarding 45-45-90 triangle can be solved using the 1:1: √2 ratio method.

## Formulas

The formulas of a 45-45-90 triangle when the lengths of its equal sides are ‘x’ units each, are given below:

## How to Find 45-45-90 Triangles

Let us solve some examples with both the possibilities to understand the concepts better.

The hypotenuse of a 45-45-90 triangle is 8√2 m. Calculate the length of its base and height.

Solution:

As we know,
Ratios of a 45-45-90 triangle side lengths = x: x: x√2, here x√2 = hypotenuse = 8√2 mm
Thus,
x√2 = 8√2 m
Squaring both sides we get,
⇒ (x√2)2 = (8√2)2 m
⇒ 2x2 = 64 x 2
⇒ x2 = 64
⇒ x = 8 m
Hence, the base and height of the given 45-45-90 triangle is 8 m each.

Solve the side lengths of a right triangle whose one angle is 45° and the hypotenuse is 6√2 cm.

Solution:

Given, one angle measures 45°, the given triangle is thus a 45-45-90 triangle.
Hence, we will use x: x: x√2 ratio of side lengths, here x√2 = hypotenuse = 6√2 cm
Dividing both sides of the equation by √2, we get
⇒ 6√2/√2 = x√2/√2
⇒ 6 = x
⇒ x = 6
Hence, the length of each side of the given 45-45-90 triangle is 6 cm.

The shorter side of an isosceles right triangle is 5√2 mm. What is the diagonal of the triangle?

Solution:

Since an isosceles right triangle is the same as a 45-45-90 triangle, here we will use x: x: x√2 ratios of side lengths to calculate the length of the hypotenuse, which is the diagonal.
Here, x = 5√2
Thus, for finding the hypotenuse,
x√2 = (5√2)√2
x√2 = 5 mm
Hence, the hypotenuse of the given 45-45-90 triangle is 5 mm.

The angle of elevation of the top of a office building from the ground is 8 cm from the base of the building is 45°. Find the elevation length of the building.

Solution:

Since, angle of elevation makes a 45-45-90 triangle with the ground,
Its side lengths = x: x: x√2, here x = 8 cm
Since,
Elevation length = hypotenuse
Substituting the value of x, we get,
x√2 = 8√2 cm
Hence, the elevation length of the top of the office building from the ground is 8√2 cm.