If *ABC* and *BDE* are two equilateral triangles such that *D* is the mid-point of *BC*, then find ar (Δ*ABC*) : ar (Δ*BDE*).

#### Solution

**Given:** (1) ΔABC is equilateral triangle.

(2) ΔBDE is equilateral triangle.

(3) D is the midpoint of BC.

**To find:** ar (Δ ABC ) : ar (ΔBDE)

PROOF : Let us draw the figure as per the instruction given in the question.

We know that area of equilateral triangle = `sqrt(3)/4 xx a^2`, where *a* is the side of the triangle.

Let us assume that length of BC is *a* cm.

This means that length of BD is `a/2` cm, Since D is the midpoint of BC.

∴ area of equilateral Δ ABC =`sqrt(3)/4 xx a^2` ------(1)

area of equilateral ΔBDE = `sqrt(3)/4 xx (a/2)^2` ------(2)

Now, ar(ΔABC) : ar(ΔBDE) =` sqrt(3)/4 xx a^2 : sqrt(3)/4 xx (a/2)^2` (from 1 and 2)

= 4 : 1

Hence we get the result ar(ΔABC) : ar(ΔBDE) = **4 : 1**