[Maths Class Notes] on Isosceles Right Triangle Pdf for Exam

A triangle comprises three sides which make three angles with each other. These angles sum to 180°.

There are Different Types of Triangle

• Equilateral Triangle

• Isosceles Triangle

• Right-Angled Triangle

• Scalene Triangle

In this article we are going to focus on definition, area, perimeter and some solved examples on   Right angled isosceles Triangle.

This triangle fulfills all the properties of the Right-angle Triangle and Isosceles Triangle. Before learning about Isosceles Right Triangle, Let us go through the properties of Right and Isosceles Triangle.

Right-Angled Triangle

A Right-angled triangle is a triangle in which one of the angles is exactly 90 degrees and the remaining other two angles sums to another 90 degrees. Since the sum of all three angles measures 180 degrees.

The two perpendicular sides of the right angle triangle are called the legs and the longest side opposite the right angle is called the hypotenuse of the triangle. You may be wondering can a Right triangle also be an isosceles triangle? Yes, a Right angle triangle can be an isosceles and scalene triangle but it can never be an equilateral triangle.

Isosceles Triangle 

An Isosceles triangle is a triangle in which at least two sides are equal. Since the two legs of the triangle are equal, which makes the corresponding angles equal to each other. Can an isosceles triangle be the right angle or scalene triangle? Yes, an isosceles can be right angle and scalene triangle. 

Definition of Isosceles Right Triangle

The Isosceles Right Triangle has one of the angles exactly 90 degrees and two sides, which are equal to each other. Since the two sides are equal which makes the corresponding angle congruent. Thus, in an isosceles right triangle two sides are congruent and the corresponding angles will be 45 degree each which sums to 90 degree. So the sum of three angles of the triangle will be 180 degrees.

Isosceles Right Triangle Formula

Pythagorean Theorem is the most important formula for any right angle triangle. Pythagorean Theorem states that the square of the hypotenuse of a triangle is equal to the sum of the square of the other two sides of the Right angle triangle. As per Isosceles right triangle the other two legs are congruent, so their length will be the same “S” and let the hypotenuse measure “H”.

Then the formula for isosceles right triangle will be:

(Hypotenuse)² = (Side)² + (Side)²

H² = S² + S²

H² = 2S²

Area of Isosceles Right Triangle

The general formula for finding out the area of a right angled triangle is (1/2xBxH), where H is the height of the triangle and B is the base of the triangle.

In an isosceles right triangle the length of two sides of the triangle are equal. Let us assume both sides measure “S” then the formula can be altered according to the isosceles right triangle.

AREA(A)= ½(SxS)

A=[frac{1}{2}times S^2]

So the area of an Isosceles Right Triangle = [frac{S^2}{2}] square units.

Perimeter of Isosceles Right Triangle

The general formula for finding out the area of any given triangle is the sum of all its sides. Thus, a triangle with side length X, Y and Z the perimeter would be:

Perimeter of a triangle = X+Y+Z units.

In the Isosceles Right Triangle the adjacent sides are equal to each other, let us assume sides “S” and hypotenuse “H”.

Thus the perimeter of an isosceles right triangle would be:

PERIMETER (P) = H+S+S.

Therefore, the perimeter of an isosceles right triangle P is H + 2S Units.

Examples on Isosceles Right Triangle 

Question1. Find the hypotenuses of an isosceles right triangle whose side is 6 cm.

Solution:

Given:

Length of the side, S = 6 cm

We know that, H² = 2S²

Substitute the value of “S” in H² = 2S² 

H² = 2(6)²

H²= 144

H = [sqrt{144}]

H = 12

Therefore, H = 12 

Therefore, the length of the Hypotenuse is 12 cm

Question 2. Find the area and perimeter of an isosceles right triangle whose hypotenuse side is 12 cm.

Solution:

Given:

Length of the hypotenuse side, H = 12 cm

We know that, H² = 2S²

Substitute the value of “H” in H2 = 2S2 

12² = 2S²

144= 2S²

S² = 144/2

S² = 72

Therefore, S = [sqrt 72] = [6sqrt{2}] cm

Therefore, the length of the congruent legs is [6sqrt{2}] cm

The area of an isosceles right triangle, A = [frac{S^2}{2}]

A = [frac{(6sqrt{2})^2}{2}]

A = [frac{(36times 2)}{2}]

A = 36

Therefore, the area of an isosceles right triangle is 36 cm²

The perimeter of an isosceles right triangle, P = H+ 2S units

P = 12 + 2[(6sqrt{2})]

P = 12 + 12[sqrt{2}]

Substitute [sqrt{2}] = 1.414

P = 12 + 12(1.414)

P = 12 + 13.14

P=25.14

Therefore, the perimeter of an isosceles right triangle is 25.14 cm

Isosceles Right Triangle Properties

The Right Isosceles Triangle follows features similar to the Isosceles Triangle. Let’s look
at a list of structures followed by an Isosceles Right Triangle:

• It has one angle equal to 90º which is the Right Angle.

• The legs of the Right Isosceles Triangle are perpendicular to each other also known as the base and height.

• The other two angles of the Right Isosceles Right Triangle are connected and measure 45 ° each.

• The sum of all the inner angles is equal to 180 °.

• The altitude drawn at Right angles is the perpendicular bisector of the hypotenuse (opposite side).

• The area of ​​the Right Isosceles Triangle is given as (1/2) × Base × Height of square units.

Why are Isosceles Triangles important?

The Isosceles Triangle is a Triangle with at least two (equal) lengths. If all three sides are equal, the Triangle is also equal. Isosceles Triangles are very useful in determining unknown angles.

How is the Isosceles Triangle used in real life?

A very popular example of an Isosceles Triangle in real life is a piece of pizza, a pair of earrings. The equal sides of an Isosceles Triangle are known as legs. The third and unequal aspect of the Isosceles Triangle is known as the base.

What is special about the Right Isosceles Triangle?

The Right Isosceles Triangle has the features of both the Isosceles and the Right Triangle. It has two equal sides, two equal angles, and one Right angle. (The Right angle may not be one of the same angles or the total angle may exceed 180 °.)