Isosceles Right Triangle: Formulas, Pythagoras Theorem and Area

Isosceles right triangle is a two dimensional three sided figure in which one angle measures 90°, and the other two angles measure 45° each. In an isosceles right triangle, Hypotenuse is given by formula H=B\(\sqrt{2}\), the area is given by B²/2, and perimeter is given by 2B+H. [H-hypotenuse, B-any of the other two sides].

Key Takeaways: Triangle, Isosceles triangle, Isosceles right triangle, Hypotenuse, Area, Perimeter.

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Triangle

In geometry, a triangle is a two-dimensional figure with three sides and three angles.

The summation of the three angles of any triangle is always 180°.

The sum of the two sides of any triangle (in isosceles right triangle too) is always greater than the third side.

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Types of Triangles

According to the measure of angles in a triangle, the triangles can be: 

• Acute 
• Obtuse 
• Right

According to the length of sides in a triangle, triangles can be: 

• Equilateral 
• Isosceles 
• Scalene

According to both angles and sides in a triangle, the triangle can be:

• Right equilateral 
• Right isosceles 
• Obtuse Scalene

Also Read:

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Isosceles triangle

A triangle in which the length of two of its sides is equal is called an isosceles triangle. Since the length of two sides of the isosceles triangle is equal, the corresponding angles are equal to each other too. 

Here, side AC=BC; and angle ∠CAB= ∠CBA

The unequal side (AB) of an isosceles triangle is usually referred to as the 'base' of the triangle.

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Right triangle

Right angle triangle is a triangle in which one angle is 90°. The longest side of the right triangle is the hypotenuse and the other two sides are height and the base. The opposite side (BC) of the right angle, whose ends are touched by arms of the angle (AB & AC), is the hypotenuse. 

AC ⊥ AB

∠A=90º

Also Read:

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Isosceles right triangle

A triangle in which one angle measures 90°, and the other two angles measure 45° each is an isosceles right triangle. In an isosceles right triangle (figure below), ∠A and ∠C measure 45° each, and ∠B measures 90°. The sides AB and AC are equal. This type of triangle is also known as a 45-90-45 triangle

AC, the side opposite of ∠B, is the hypotenuse. And AB or AC can be taken as height or base

Also Read: Quadrilateral

Formulas

Given below are the formulas to construct a triangle which includes:

1. Pythagoras theorem
2. Area
3. Perimeter

Pythagoras Theorem

Pythagoras theorem, which applies to any right-angle triangle, also applies to isosceles right triangles. 

In an isosceles right triangle, the length of base (BC) is equal to length of height (AB). If base (BC) is taken as ‘B’, then AB=BC=’B’. Let the hypotenuse be taken as ‘H’

Pythagoras theorem states that the square of the hypotenuse of a right triangle is equal to the sum of the square of the other two sides.

(Hypotenuse)² = (Side)² + (Side)²

H²=B²+B²

H²=2B²
H=B\(\sqrt{2}\)

Area

The area of a triangle is half of the base times height.

Area = ½ × b × h              

Here, b-base; h-height

This applies to right isosceles triangles also. 

As stated above, in an isosceles right-triangle the length of base (BC) is equal to length of height (AB). If, base (BC) is taken as ‘B’, then AB=BC=’B’

Area of right isosceles triangle= ½ ×B×B

= B²/2

Perimeter

The Sum of all sides of a triangle is the perimeter of that triangle. 

In the right isosceles triangle, since two sides (Base BC and Height AB) are same and taken as ‘B’ each. And the hypotenuse is taken as ‘H’. 

The perimeter= AB+BC+AC

= B+B+H

=2B+H

Also Read: 

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Things to Remember

• Triangles are polygons with the least number of sides; since any figure with two sides only forms an angle.
• In the right triangle two sides are perpendicular to each other; in the right isosceles triangle these two sides are of equal length.
• The difference between any two sides is smaller than the 
• third side-AB-BC<AC
• The concept of right isosceles triangles is used by architects. Even Egyptians used these to design pyramids.

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