Question

ABC is a triangle right angled at C and AC= √3 BC. prove that angle ABC = 60°

Answer 1

Draw a median CD to AB from C.

Now, by Pythagoras theorem in ABC, AB=2*BC.

Thus, BD= half of AB = BC.

But, the point of bisection of the hypotenuse is equidistant from all sides.

Thus, CD=BD=AD.

That implifies, CD=BD=BC.

Thus, CDB is an equilateral triangle.

Therefore, angle DBC = angle ABC = 60 degrees.

Answer 2

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°.