Question

If ΔABC ∼ ΔDEF and the heights of ΔABC and ΔDEF are in the ratio of 8 ∶ 13. By what percentage is the area of ΔABC less than the area of ΔDEF? 

1. 60.45%
2. 50.26%
3. 70.11%
4. 62.13%

Answer 1

Correct Answer - Option 4 : 62.13%

Given:

ΔABC ∼ ΔDEF

The heights of ΔABC and ΔDEF are in the ratio of 8 ∶ 13.

Concepts used:

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

Calculation:

ΔABC ∼ ΔDEF

⇒ ar(ABC)/ar(DEF) = [Height of ΔABC/height of ΔDEF]²

⇒ ar(ABC)/ar(DEF) = (8/13)²

⇒ ar(ABC)/ar(DEF) = 64/169

Let the ar(ABC) be 64x and ar(DEF) be 169x.

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

⇒ [(169x – 64x)/169x] × 100

⇒ 62.13%

∴ Area of ΔABC is less than the area of ΔDEF by 62.13%.