Question

On sides AB, BC and AC of triangle ABC, equilateral triangle ABC', BCA' and CAB' are drawn. Prove that AA' = BB' = CC'​

Answer 1

Answer:

AA' = BB' = CC'

Step-by-step explanation:

Given,

On sides, AB, BC, and AC of triangle ABC, equilateral triangle ABC', BCA', and CAB' are drawn.

To Find,

Prove that AA' = BB' = CC'​

Solution,

In ΔACA' and triangle B'CB;

AC = B'C ....( Sides of an equilateral triangle ACB' )

A'C = BC......( Sides of an equilateral triangle BCA')

Therefore,

∠ACA' = ∠B'CB .......[ ∠ACA' = ∠ACB + ∠BCA' = ∠ACB + 60

                                   ∠B'CB = ∠ACB + ∠B'CA = ∠ACB + 60 ]

Therefore by SAS congruency:

ΔACA' ≅ ΔB'CB

and thus, by CPCT, AA' = BB'...........(1)

similarly, we can show that AA' = CC'..............(2)

therefore,

AA' = BB' = CC'

Hence Proved.