CAT 2018

Slot 1

QA

Question & Solution

1. Concept Used

• Topic: Geometry – Rectangle Inscribed in a Circle (Thales' Theorem)
• Formula: For a rectangle inscribed in a circle, the diagonal of the rectangle equals the diameter of the circle. Using the Pythagorean theorem: $$ l^2 + b^2 = d^2 = (2r)^2 $$

2. Calculation

The radius of the circle is given as (r = 13) cm, so the diameter (which equals the diagonal of the rectangle) is (d = 2 \times 13 = 26) cm.

Since any rectangle inscribed in a circle has its diagonal equal to the diameter of the circle, and since all angles of a rectangle are (90°), by the Pythagorean theorem applied to the right triangle formed by the diagonal:

$$l^2 + b^2 = 26^2 = 676$$

Now we verify each option:

Option A (24, 10): (24^2 + 10^2 = 576 + 100 = 676) ✅ — This satisfies the condition.

Option B (25, 9): (25^2 + 9^2 = 625 + 81 = 706 eq 676) ❌

Option C (25, 10): (25^2 + 10^2 = 625 + 100 = 725 eq 676) ❌

Option D (24, 12): (24^2 + 12^2 = 576 + 144 = 720 eq 676) ❌

Only Option A satisfies (l^2 + b^2 = 676).

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3. Solution

Answer = Option A ✅

The only valid pair of length and breadth is (24, 10), since (24^2 + 10^2 = 676 = 26^2), confirming the diagonal equals the diameter of the circle.