Question

If ΔDEF ∼ ΔPQR and the perimeter of ΔDEF and ΔPQR is in the ratio of 9 ∶ 10. By what percentage is the area of ΔDEF less than the area of ΔPQR?
1. 25%
2. 40%
3. 20%
4. 19%

Answer 1

Correct Answer - Option 4 : 19%

Given:

ΔDEF ∼ ΔPQR

The perimeter of ΔDEF and ΔPQR is in the ratio of 9 ∶ 10.

Concepts used:

The ratio of area of similar triangles is equal to the square of the ratio of perimeter or sides of corresponding triangles.

Calculation:

ΔDEF ∼ ΔPQR

⇒ ar(DEF)/ar(PQR) = [perimeter(ΔDEF)/perimeter(ΔPQR)]²

⇒ ar(DEF)/ar(PQR) = (9/10) ²

⇒ ar(DEF)/ar(PQR) = 81/100

Let the ar(DEF) be 81x and ar(PQR) be 100x.

⇒ Percentage by which ar(DEF) is less than ar(PQR) = [(ar(PQR) – ar(DEF))/ar(DEF)] × 100

⇒ [(100x – 81x)/100x] × 100 = 19%

∴ Area of ΔDEF is less than the area of ΔPQR by 19%.