The question is from the topic circles and polygons.It discusses about the ratio of circumference of circle to the polygon that is inscribed in it. CAT Geometry questions are heavily tested in CAT exam. Make sure you master Geometry problems.

Question 35: There are 2 concentric circles, one big and one small. A square ABCD is inscribed inside the big circle while the same square circumscribes the small circle. The square touches the small circle at points P, Q, R and S. Determine the ratio of circumference of big circle to the polygon PQRS.

- π : 2
- 2 : π
- 2 : √2
- π : √2

π : 2

Let the side of the square ABCD be 2a.

A line from the centre will bisect the chords AB, BC, CD and DA

Therefore AP = PB = a. Similarly for other sides too.

From the above we can conclude that PQ = QR = RS = SP. Therefore PQRS is also a square.

Radius of big circle = diagonal of the square APOS = a√2

Circumference = 2π * a√2

Now we have to find the side of square PQRS.

We know that radius of small circle = side of the square APOS = a

Consider triangle PSO. PS^{2} = OP^{2} + OS^{2}. Therefore, PS = a√2

Circumference of square with side a√2 = 4a√2

Ratio = 2π * a√2 : 4a√2

=> π : 2

The question is **"Determine the ratio of circumference of big circle to the polygon PQRS."**

Choice A is the correct answer.

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