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