Question

∆ABC and ∆BDE are two equilateral triangles such that D is the mid-point of BC. The ratio of the areas of triangles ABC and BDE is ______.

Options

• 2 : 1

• 1 : 2

• 4 : 1

• 1 : 4

Answer 1

∆ABC and ∆BDE are two equilateral triangles such that D is the mid-point of BC. The ratio of the areas of triangles ABC and BDE is 4 : 1.

Explanation:

Given: ΔABC and ΔBDE are two equilateral triangles such that D is the midpoint of BC.

To find: Ratio of areas of ΔABC and ΔBDE.

ΔABC and ΔBDE are equilateral triangles; hence, they are similar triangles.

Since D is the midpoint of BC, BD = DC.

We know that the ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding sides.

`\text{ar(Δ ABC)}/\text{ar(Δ BDE)} = ((BC)/(BD))^2`

`\text{ar(Δ ABC)}/\text{ar(Δ BDE)} = ((BD+DC)/(BD))^2`  ...[D is the midpoint of BC]

`\text{ar(Δ ABC)}/\text{ar(Δ BDE)} = ((BD+DC)/(BD))^2`

`\text{ar(Δ ABC)}/\text{ar(Δ BDE)} = ((2BD)/(BD))^2`

`\text{ar(Δ ABC)}/\text{ar(Δ BDE)} = 4/1`

Hence, the ratio of the areas of triangle ABC and BDE is 4 : 1.