Question

abc and dbf are equilateral triangles such that d is mid-point of bc. What is the ratio of the area of bae to the area of bef?

Answer 1

abc and dbf are equilateral triangles such that d is mid-point of bc. The ratio of the area of bae to the area of bef is 1:4

Two equilateral triangle are similar.

Therefore abc ≈ dbf

∴ We have,

df/ac = 12/6 = 2

ef/bc = 12/6 = 2

df/ab = 12/6 = 2

Hence using SSS theorem, we have,

abc ~ dbf

We know that, the ratio of area of triangles is equal to the ratio  of square of the corresponding sides

Therefore, we have,

ar of Δabc / ar of Δbde = bc² / bd²

as, bd = 1/2 bc

we have,

ar of Δabc / ar of Δbde = bc² / (1/2bc²)

= bc² / (bc²/4)

= 4bc² / bc²

= 4 / 1

the ratio of the area of bae to the area of bef is 1:4