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

Solution:

Given, a semicircle is drawn on the hypotenuse of a right angled triangle.

Also, semicircles are drawn on the other two sides of the triangle.

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

Let ABC be a right triangle with B at right angle.

Let AB = y and BC = x

A semicircle is drawn on the hypotenuse with AC as the diameter.

Two semicircles are drawn on the other two sides with AB and BC as the diameter.

Let A₁ be the area of the semicircle drawn on the hypotenuse

A₂ and A₃ the area of the semicircles drawn on the other two sides

By pythagoras theorem,

AC² = AB² + BC²

AC² = y² + x²

Taking square root,

AC = √(x² + y²)

We know that, area of semicircle = πr²/2

Area of semicircle drawn on side AC = π(AC/2)² = πAC²/8

A₁ = π/8(√(x² + y²))²

= π(x² + y²)/8 --------------------- (1)

Area of semicircle drawn on AB = π(AB/2)² = πAB²/8

A₂ = πy²/8

Area of semicircle drawn on BC = π(BC/2)² = πBC²/8

A₃ = πx²/8

Sum of areas of semicircle drawn on AB and BC = A₂ + A₃

= πy²/8 + πx²/8

= π(x² + y²)/8 ------------------ (2)

From (1) and (2),

A₁ = A₂ + A₃

Therefore, the area of a semicircle drawn on the hypotenuse is equal to the sum of the area of semicircles drawn on AB and BC.

✦ Try This: If the sum of the hypotenuse and a side of a right angled triangle is given, show that the area of the triangle is maximum when the angle between them is π/3.

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

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NCERT Exemplar Class 10 Maths Exercise 6.4 Problem 17

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

Summary:

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

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