Here sir the question is not assuming is a perfect triangle. it is a deduction based on the information in the stem

What information in the stem leads us to figuring out it is an equilateral?

We are told that triangle \(ABC\) is contained within a circle with center \(C\) and points \(A\) and \(B\) lie on the circle.

We are also given that area of the circle is \(25\pi\).

We know that area of triangle \(= \pi \times radius^2\). Thus we can say: \(\pi \times radius^2 = 25 \times \pi\)

\(⇒ \pi \times radius^2 = 25 \times \pi\)

\(⇒ radius^2=25\)

\(⇒ radius=\sqrt{25}\)

\(⇒ radius=5\)

We are given that \(\angle ACB = 60^o\)

After all this information, we can draw:




Since \(AC=CB\), the angles opposite to them i.e., \(\angle A\) and \(\angle B\) will also be same

Thus we can say: \(\angle A+\angle B +\angle C = 180^o\)

\(⇒ \angle A + \angle A + 60 = 180\)

\(⇒ 2\times \angle A = 120\)

\(⇒ \angle A = 60^o\)

Thus \(\angle A = \angle B = \angle C = 60^o\)

So, triangle ABC is an equilateral triangle. So the possible value of \(AB = 5\).

Hence the right answer is Option C.

For all the students above: we have to infer information from the stem NOT always we can assume information from the stem!!!