Question

ABC is a triangle in which BE and CF are, respectively, the perpendiculars to the sides AC and AB. If BE = CF, prove that ΔABC is isosceles  

Answer 1

Given that ABC is a triangle in which BE and CF are perpendicular to the sides AC and AB respectively such that BE = CF   

We have to prove that ΔABC is isosceles
Now, consider ΔBCF and ΔCBE, 

We have 

∠BFC=CEB=90°                 [Given] 

BC=CB                             [Common side] 

And CF=BE                        [Given]  

So, by RHS congruence criterion, we have ΔBFC≅CEB 

Now, 

∠FBC=∠EBC   [∵ Incongruent triangles corresponding parts                                                   are equal] 

⇒ ∠ABC=∠ACB 

⇒ AC=AB    [ ∵Opposite sides to equal angles are equal in a                                triangle]

∴ ΔABC is isosceles