# Right Isosceles Triangle

Sub Topics
A triangle is either classified by side attributes or by angle attributes.

 Classification of triangles according to side attributes Scalene triangle No two sides of the triangle are congruent. Isosceles triangle At least two sides of the triangle are congruent Equilateral triangle All the sides of the triangle are congruent Note the equilateral triangle is a special isosceles triangle.

 Classification of triangles according to angle attributes Acute triangle All the three angles are acute Obtuse triangle One angle in the triangle is obtuse. Right triangle One angle in the triangle is a right angle All the triangles have at least two acute angles. Hence the classification is done on the basis of the third angle.

Hypotenuse and Legs of a right triangle

Special names are given to the sides of a right triangle based on their positions relative to the right angle.

Hypotenuse is the side opposite to the right angle.

In the given triangle side BC is the hypotenuse, as it is opposite to the right angle ‘A’.

The sides containing the right angle are called the legs of the right triangle. Sides AC and AB are the legs of the given right triangle ABC.

Hypotenuse is the longest side in a right triangle as it is opposite to the largest angle.

## What is a Right Isosceles Triangle

Right isosceles and isosceles right triangles refer to the same type, meaning right triangle with two congruent sides.  When we say two sides are congruent, which two sides do we mean? Hypotenuse being the longest side in a right triangle, a leg can never be congruent to the hypotenuse. Hence in a right isosceles triangle the legs containing the right angle must be congruent.
Right Isosceles Triangle Definition:

A right isosceles triangle is a right triangle in which the legs containing the right angle are congruent.

The congruent sides in any isosceles triangles are also known as legs.

## Isosceles Right Triangle Formula

In an isosceles triangle in addition to two sides being congruent, the base angles, that is the angles formed on the third side are also congruent. That makes the angles made by the legs on the hypotenuse are congruent. Since the vertex angle is a right angle, the sum of the measures of the base angles in a right isosceles triangle is 90o.

Hence the measure of each of the non right angle is 450.

The measure of each of the non right angles in a right isosceles triangle is 45o.

How are the legs and hypotenuse related in a right isosceles triangle?

The Pythagorean Theorem also known as right triangle formula gives the equation satisfied by the sides of all right triangles

In a right triangle sum of the squares on the legs is equal to the square on the hypotenuse.

c2 = a2 + b2

where 'a' and 'b' are the lengths of the legs and 'c' is the length of the hypotenuse.

In a right triangle sum of the squares on the legs is equal to the square on the hypotenuse.

c2 = a2 + b2 = 2a2 → c= $\sqrt{2a^2}$

where ‘a’ and b’ are the lengths of the legs and c is the length of the hypotenuse.

c=$\sqrt{2a}$

Isosceles right triangle formula:

The right isosceles triangle is also known as a 45, 45, 90 special right triangle indicating the measures of the angles.

The ratio of sides opposite to the respective angles in a right isosceles triangle is 1:1:$\sqrt{2}$

The example given demonstrates the configuration of a right isosceles triangle both in terms of angles and side lengths

Length of the hypotenuse $\sqrt{2}$ x length of the leg.

## Can a Right Triangle be Isosceles

The classification can also be done combining both the side and angle attributes. But not all side types can be combined with each of the angle type. For example an equilateral triangle can only be an acute triangle, and it is not possible to find an equilateral right or right equilateral triangle. But an isosceles triangle can have all the three types of classification done by angles. It is possible to define a triangle as isosceles acute, isosceles obtuse or isosceles right triangle. Same way a right triangle can be a right scalene triangle or a right isosceles triangle.

## Formula for the Diagonal of a Square

The formula for the length of the diagonal of a square in terms of the side length can be derived using the isosceles right triangle formula.

The diagonal divides a square into two congruent right isosceles triangles. As shown in the diagram, the two adjacent sides of the square are the legs and the diagonal is the hypotenuse. If 'a' is the side of the square, then by the property of right isosceles triangle, Length of the diagonal of a square=$\sqrt{2a}$

## Area of an Right Isosceles Triangle

The general formula for the area of a triangle is,

Area $\triangle$=$\frac{1}{2}$. bh  where b is the base and h the corresponding height of the triangle.

In a right triangle, the two legs can be taken as the base and the altitude.  Hence the area formula gets simplified for a right triangle as

Area of a right triangle=$\frac{1}{2}$x Leg 1 x Leg 2.

Further in a right isosceles triangle both the legs are of equal length. The formula for area can be hence simplified further.

Area of a right isosceles triangle  $\frac{1}{a^2}$.

Where 'a' is the length of each leg.

Example for finding the area of a right isosceles triangle

Example 1: Find the area of a right isosceles triangle whose legs measure a length 12 cm each.

Length of the leg = a = 12 cm

Area of the right isosceles triangle =$\frac{a^2}{2}$=$\frac{12^2}{2}$=$\frac{144}{2}$ = 72cm2

Example 2: If the area of a right isosceles triangle is 18 cm2, find the legs and hypotenuse of the right triangle.

Area of the right isosceles triangle=$\frac{a^2}{18}$=18 → a2=36 → a = 6cm.

Hypotenuse of the right triangle  $6\sqrt{2a^2}$cm  and  each leg = 6 cm.

## Circumcircle of a Right Isosceles Triangle

In general the altitude drawn from the vertex of an isosceles triangle bisects the opposite side. In the case of a right isosceles triangle side opposite to the right angle is the hypotenuse. The perpendicular bisectors of the legs meet at this point. Hence the circumcenter falls on the hypotenuse. The diagram given here shows the right isosceles triangle inscribed in the circumcircle, with the midpoint of the hypotenuse as the circumcenter.