45-45-90 Triangle / Isosceles Right Triangle

Last updated on 01 February 2025

Table of contents

Definition of 45-45-90 triangle
Properties of the 45-45-90 Triangle
Perimeter of Isosceles Right Triangle
Area of Isosceles Right Triangle
Altitude of Isosceles Right Triangle
Summary

The 45-45-90 triangle is a special type of right-angled triangle that holds distinct geometric properties. Its name originates from the angles it contains: two 45° angles and one 90° angle. This unique configuration gives rise to specific ratios and characteristics that make the 45-45-90 triangle a fascinating topic in geometry. Its simple and consistent ratios make the 45-45-90 triangle a valuable tool in problem-solving. Students often encounter it in geometry and trigonometry exercises, and understanding its properties aids in solving related mathematical problems. Let's explore the properties, relationships, and applications of the 45-45-90 triangle.

Definition of 45-45-90 triangle

A 45-45-90 triangle is a special type of right-angled triangle where the measures of the three angles are 45°, 45° and 90°. In other words, it is an isosceles right triangle, meaning that two of its angles are equal, and the third angle is a right angle.

△ LMN

in the figure above,

∠ L

and

∠ N

measure 45° and

∠ M

is right angle. Hence it is 45-45-90 triangle or an isosceles right triangle. Hence sides opposite to 45° angle are equal.

Properties of the 45-45-90 Triangle

Angle Measures

The 45-45-90 triangle consists of two equal angles measuring 45° and one right angle measuring 90°. These angles always add up to 180°, as in any triangle. With these angle measures, we can say that the three angles are in the ratio 1:1:2.

Isosceles Nature

An isosceles triangle has two sides of equal length. In the 45-45-90 triangle, the sides opposite to 45° angles are called the legs of the triangle and have equal lengths. Therefore, the 45-45-90 triangle is an isosceles triangle.

Side Length Ratios

The most distinctive property of the 45-45-90 triangle is the relationship between its side lengths. The three sides of such a triangle are in the ratio

1 : 1 : \sqrt{2}

. If the measure of the legs of such a triangle is

a

each, the length of the hypotenuse (the side opposite the right angle) is

\sqrt{2}

times the length of the legs. Mathematically, this can be expressed as:

Hypotenuse = a \sqrt{2}

In the figure above,

△ PQR

is a 45-45-90 triangle.

∠ P = ∠ R = 45 °

∠ Q = 90 °

PQ and QR are the legs of the triangle and PR is hypotenuse.

PQ = QR = a

By Pythagorean theorem,

\begin{array}{lrcl} ∵ & {PR}^{2} & = & {PQ}^{2} + {QR}^{2} \\ ∴ & {PR}^{2} & = & a^{2} + a^{2} \\ ∴ & {PR}^{2} & = & 2 a^{2} \\ ∴ & PR & = & \sqrt{2} \times a \\ ∴ & PR & = & a \sqrt{2} \end{array}

Hence, $Hypotenuse = a \sqrt{2}$

Now, check the ratio of the sides of the triangle.

\begin{array}{rcl} PQ : QR : PR & = & a : a : a \sqrt{2} \\ PQ : QR : PR & = & 1 : 1 : \sqrt{2} \end{array}

This ratio can be used to find the unknown side lengths of the 45-45-90 triangle.

Understanding these properties not only aids in solving problems related to the 45-45-90 triangle but also lays the groundwork for comprehending more advanced geometric and trigonometric concepts. The unique characteristics of the 45-45-90 triangle make it a valuable tool in both theoretical mathematics and practical applications.

Example 1: Find the length of the sides

XY

and

YZ

△ XYZ

in the figure below, if

c = 5 \sqrt{2} cm

Given: In

△ XYZ

XY = YZ = a

XZ = c = 5 \sqrt{2} cm

∠ X = ∠ Z = 45 °

∠ Y = 90 °

To find: Length of the sides XY and YZ

Solution: The given triangle is 45-45-90 triangle.

\begin{array}{lrcl} ∵ & XY : YZ : XZ & = & a : a : a \sqrt{2} \\ ∴ & XZ & = & a \sqrt{2} \\ ∴ & 5 \sqrt{2} & = & a \sqrt{2} \\ ∴ & \frac{5 \sqrt{2}}{\sqrt{2}} & = & a \\ ∴ & 5 & = & a \\ ∴ & a & = & 5 cm \end{array}

Therefore,

XY = YZ = 5 cm

Example 2: Find the length of the hypotenuse of the 45-45-90 triangle if its legs measure

3 \sqrt{2} cm

Given: In the given 45-45-90 triangle,

a = 3 \sqrt{2} cm

To find: Length of the hypotenuse (c)

Solution:

\begin{array}{lrcl} ∵ & a : a : c & = & a : a : a \sqrt{2} \\ ∴ & c & = & a \sqrt{2} \\ ∴ & c & = & 3 \sqrt{2} \times \sqrt{2} \\ ∴ & c & = & 3 \times {(\sqrt{2})}^{2} \\ ∴ & c & = & 3 \times 2 \\ ∴ & c & = & 6 cm \end{array}

Therefore, the length of the hypotenuse is 6 cm.

Perimeter of Isosceles Right Triangle

The perimeter of a triangle is the sum of the lengths of its all three sides. If the length of the equal sides and hypotenuse are given for an isosceles right triangle, the formula to calculate the perimeter is expressed as:

P = 2 a + b

where,

a

is the length of one of the two equal sides and

b

is the length of the hypotenuse (unequal side) of the isosceles triangle.

Example: Find the perimeter of an isosceles right triangle if length of the two equal sides is 4 cm and length of the hypotenuse is 6 cm.

Given: $a = 4 cm$ , $b = 6 cm$

To find: Perimeter of the given isosceles right triangle

Solution: $\begin{array}{lrcl} ∵ & P & = & 2 a + b \\ ∴ & P & = & 2 \times 4 + 6 \\ ∴ & P & = & 8 + 6 \\ ∴ & P & = & 14 cm \end{array}$

Therefore, the perimeter of the given isosceles right triangle is

14 cm

Area of Isosceles Right Triangle

The area of a triangle is the measure of the amount of space it occupies in a two-dimensional plane. The area of an isosceles right triangle can be calculated using the formula below:

Area = \frac{1}{2} \times a^{2}

where

a

is the length of one of the two equal sides of an isosceles right triangle.

Example: Find the area of an isosceles right triangle if the length of it's two equal sides is 4 cm.

Given: $a = 4 cm$

To find: Area of the given isosceles right triangle

Solution: $\begin{array}{lrcl} ∵ & A & = & \frac{1}{2} \times a^{2} \\ ∴ & A & = & \frac{1}{2} \times 4^{2} \\ ∴ & A & = & \frac{1}{2} \times 16 \\ ∴ & A & = & 8 {cm}^{2} \end{array}$

Therefore, the area of the given isosceles right triangle is

8 {cm}^{2}

Altitude of Isosceles Right Triangle

The formula for an altitude or height of an isosceles right triangle is:

h = \sqrt{a^{2} - \frac{b^{2}}{4}}

where

$a$ = length of one of the two equal sides
$b$ = length of the hypotenuse (the unequal side)

Example: Find altitude of an isosceles right triangle given the length of equal sides is

7 cm

and the hypotenuse is

10 cm

Given: $a = 7 cm$ , $b = 10 cm$

\begin{array}{lrcl} ∵ & h & = & \sqrt{a^{2} - \frac{b^{2}}{4}} \\ ∴ & h & = & \sqrt{7^{2} - \frac{10^{2}}{4}} \\ ∴ & h & = & \sqrt{49 - \frac{100}{4}} \\ ∴ & h & = & \sqrt{49 - 25} \\ ∴ & h & = & \sqrt{24} \\ ∴ & h & = & \sqrt{4 \times 6} \\ ∴ & h & = & \sqrt{2^{2} \times 6} \\ ∴ & h & = & 2 \times \sqrt{6} \\ ∴ & h & = & 2 \sqrt{6} cm \end{array}

Therefore, the altitude of the given isosceles right triangle is

2 \sqrt{6} cm

Summary

In conclusion, the 45-45-90 triangle stands out as a distinctive geometric shape with unique properties and applications. Its isosceles nature, angle measures, and consistent side length ratios make it a valuable tool in various fields, from construction to mathematics. A solid understanding of the 45-45-90 triangle enhances problem-solving skills and provides a foundation for exploring more advanced geometric and trigonometric concepts.