Isosceles Right Triangle

Isosceles Right Triangle is the one which has:
(a) One 90 degree angle
(b) Two sides of equal length

As the name suggests Isosceles Right Triangle has features of both right triangle and Isosceles Triangle

Isosceles Right Triangle Examples
Example 01

Given above is the Right Angled Isosceles Triangle.
Notice the following features:
(a) Angle B measures 90 degree
(b) Two sides AB & BC are equal
AB = BC = 4 cms

Example 02

Triangle ABC is a Isosceles Right Triangle, where:
(a) Angle A measures exactly 90 degrees
(b) Side AB & AC are equal
AB = AC = 5 cms

Structure of Isosceles Right Triangle

Below is triangle ABC with sides AB = BC and angle B = 90

Isosceles Right Triangle consists of following components

(a) Hypotenuse
The longest side of the triangle is known as Hypotenuse
Here side AC is the hypotenuse

(b) Base
The side placed horizontally is called base

(c) Height
The line perpendicular to the base is called Height

Note
The sides Base and Height are at 90 degree with each other

Properties of Right Isosceles Triangle

(01) The side opposite to 90 degree angle is the longest
Because 90 degree is the largest angle in right isosceles triangle, its opposite will be the longest.

This longest side is also called Hypotenuse

(02) The two equal sides also have equal angles whose value is 45 degree

Given above is Right Isosceles Triangle with sides AB = BC
Since both sides are equal, the opposite angles are also equal.
Hence, \angle A =\angle C

Proof
The other two angles of Right Isosceles triangle measure 45 degree each

We know that:
\angle B = 90 degree
Let \angle A =\angle C =x

Using Angle property of Triangle
∠A + ∠B +∠C = 180
x + 90 + x = 180
2x = 90
x = 45 degree

Hence, Let \angle A =\angle C =45 degree

(03) Location of Circumcenter, Centroid and Orthocenter

All the points, circumcenter, Centroid and Orthocenter are located inside the triangle

Area of Right Isosceles Triangle

ABC is Right Isosceles Triangle where:
Base = Height = x cm

We know that:

Area=\ \frac{1}{2} \times \ base\ \times height

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

Frequently asked Question – Right Isosceles Triangle

(01) Is Right Triangle and Right Isosceles Triangle the same thing?

(02) Can Right Isosceles triangle have Obtuse angle?

(03) What is the Perimeter of Right Isosceles Triangle?

Read Solution

Perimeter can be calculated by adding all the sides

In the above right isosceles triangle, the perimeter can be calculated as:
Perimeter = x + x + a
Perimeter = 2x + a