Q25 of 56 Page 6

ABC is a triangle, right-angled at C and AC = 𢆣 BC. Prove that ∠ABC = 60°.

Given: ΔABC is right angled at C and AC = �.
 
To prove: ∠ ABC = 60°.
Proof:
Let D be the midpoint of AB. Join CD.
Now, AB2 = BC2 + AC2 = BC2 + (BC)2 = 4BC2
Therefore AB = 2BC.
Now, BD = AB = (2BC) = BC.
But, D being the midpoint of hypotenuse AB, it is equidistant from all the three vertices.
Therefore CD = BD = DA or CD = AB = BC.
Thus, BC = BD = CD,
i.e., ΔBCD is a equilateral triangle.
Hence, ∠ABC = 60°.

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