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 : ΔABC and ΔBDE are equilateral triangles. D is the point of BC.

 

To find : ar(ΔABC) : ar(ΔBDE)

 

Proof :  

ΔABC∼ΔBDE (By AAA criteria of similarity)

[all angles of the equilateral triangles are equal]

 

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

 

arΔABC/ arΔBDE = (BC/BD)²

BD = DC as D is the midpoint of BC.

 

Hence , arΔABC/ arΔBDE = ((BD + DC)/ BD)²

arΔABC/ arΔBDE = ((BD +  BD )/ BD)²

arΔABC/ arΔBDE = (2BD/BD)²

arΔABC/ arΔBDE = (2/1)²

arΔABC/ arΔBDE = 4/1

arΔABC : arΔBDE = 4 : 1

Hence, the ratio of areas of ΔABC  and ΔBDE is 4 : 1 .  

HOPE THIS ANSWER WILL HELP YOU…..

 

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