In the given figure, a circle is inscribed in a quadrilateral ABCD in which $\angle B = 90°$. If AD = 17 cm, AB = 20 cm and DS = 3 cm, then find the radius of the circle.
Generated by claude-sonnet-4-6 · 2026-06-15 10:27 · grounding rag
Model Answer
Given: ∠B = 90°, AD = 17 cm, AB = 20 cm, DS = 3 cm. Let radius = r.
Step 1: Since tangents from an external point are equal:
- DS = DR = 3 cm (tangents from D)
- AR = AD − DR = 17 − 3 = 14 cm
- AQ = AR = 14 cm (tangents from A)
- BQ = AB − AQ = 20 − 14 = 6 cm
Step 2: Since ∠B = 90°, OQBP is a square (OQ ⊥ AB, OP ⊥ BC, OQ = OP = r):
$$BQ = BP = r$$
Step 3: From Step 1, BQ = 6 cm.
$$\therefore r = 6 \text{ cm}$$
Source: Chapter 10, Section 10.3 (Theorem 10.2)
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Explanation
- Key theorem used: Tangents from an external point are equal in length (Theorem 10.2).
- At the right angle vertex B, the two radii to the points of tangency are both perpendicular to the sides, and ∠B = 90°, so OQBP is a square → BQ = BP = r.
- Work around the figure using equal tangent lengths to find each segment, then equate BQ to r.
- Examiners award marks for: correct use of equal tangents, identifying the square at B, and the final value.