Prove that the area of the equilateral triangle drawn on the hypotenuse of a right angled triangle is equal to the sum of the areas of the equilateral triangles drawn on the other two sides of the triangle

Solution:

Given, an equilateral triangle is drawn on the hypotenuse of a right triangle.

Two equilateral triangles are drawn on the other two sides of the triangle.

We have to prove that the area of the equilateral triangle drawn on the hypotenuse of a right angled triangle is equal to the sum of the areas of the equilateral triangles drawn on the other two sides of the triangle.

Let PQU be an equilateral triangle drawn with the hypotenuse PQ

Let PRS and QRT be the equilateral triangles drawn on the sides PR and RQ respectively.

Let PR = b an RQ = a

Let x₃ be the area of the equilateral triangle drawn on the hypotenuse

x₂ and x₁ be the area of the equilateral triangles drawn on the other two sides

In △PRQ,

By pythagoras theorem,

PQ² = PR² + RQ²

PQ² = b² + a²

Taking square root,

PQ = √(a² + b²)

We know that, area of the equilateral triangle = √3/4(side)²

Area of △PQU  = x₃

= √3/4(√(a² + b²))²

x₃ =√3/4(√(a² + b²))²

x₃ =√3/4 (a² + b²)------------------ (1)

Area of △QRT= x₁

x₁= √3/4(PQ)²

x₁ = √3/4(a²)

Area of △PRS = x₂

= √3/4(PR)²

x₂ = √3/4(b²)

Sum of the area of the equilateral triangles drawn on side RQ and PR = x₁ + x₂

= √3/4(a²) + √3/4(b²)

= √3/4(a² + b²) ----------------------- (2)

From (1) and (2),

x₃ = x₁ + x₂

Therefore, the area of the equilateral triangle drawn on the hypotenuse of a right angled triangle is equal to the sum of the areas of the equilateral triangles drawn on the other two sides of the triangle.

✦ Try This: In a right angled triangle with sides a and b and hypotenuse c, the altitude drawn on the hypotenuse is x. Prove that ab = cx

☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 6

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NCERT Exemplar Class 10 Maths Exercise 6.4 Problem 18

Prove that the area of the equilateral triangle drawn on the hypotenuse of a right angled triangle is equal to the sum of the areas of the equilateral triangles drawn on the other two sides of the triangle

Summary:

It is proven that the area of the equilateral triangle drawn on the hypotenuse of a right angled triangle is equal to the sum of the areas of the equilateral triangles drawn on the other two sides of the triangle

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