Question

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

Answer 1

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