**ABC is a triangle, right-angled at C and Ac = √3 BC. Prove that ABC = 60°.**

###### Answers:

Given: △ABC is right angled at C and AC = √3BC.

To prove: ∠ABC = 60°.

Proof:

Let D be the midpoint of AB. Join CD.

Now, AB2 = BC2 + AC2 = BC2 + (√3BC)2 = 4BC2

Therefore AB = 2BC.

Now, BD = 1/2 AB = 1/2(2BC) = BC.

But, D being the midpoint of hypotenuse AB, it is equidistant from all the

three vertices.

Therefore CD = BD = DA or CD = 1/2 AB = BC.

Thus, BC = 80 = CD,

i.e., △BCD is a equilateral triangle.

Hence, ∠ABC = 60°.