Mathematics — Standard (041) — CBSE Class 10 board question

Proof:

Let the circle (radius 10 cm) touch sides AB, BC, CD, and DA at points P, Q, R, and S respectively.

Since tangents drawn from an external point to a circle are equal (Theorem 10.2):

• From A: AP = AS
• From B: BP = BQ
• From C: CQ = CR
• From D: DR = DS

Since the circle touches all four sides and has radius 10 cm, the distance between each pair of opposite sides equals 2r = 20 cm.

Thus AB = CD = 20 cm and BC = AD = 20 cm.

Since all four sides are equal and each angle of a rectangle is 90°, ABCD is a square.

Perimeter = 4 × 20 = 80 cm

Source: Chapter 10, Section 10.3 (Theorem 10.2)

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• Key theorem used: Tangents from an external point are equal (Theorem 10.2).
• For a rectangle circumscribing a circle, the diameter equals the width, so each side = 2r = 20 cm. All sides being equal + all angles 90° → square.
• Examiners expect you to state Theorem 10.2, apply it at each vertex, conclude all sides equal, then compute perimeter. Don't skip the equal-tangent step at each corner.