Class 10 Tangent Lengths and Incircles — practice questions with answers
Geometry · 5 sample questions from the school syllabus. Try each one, then open the answer and the worked solution.
1. From a point P, two tangents PA and PB are drawn to a circle. If PA = 7 cm, what is PB?
- A3.5 cm
- B7 cm
- C14 cm
- DIt cannot be found
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Answer: B) 7 cm
The lengths of the two tangents drawn from an external point to a circle are equal, so PB = PA = 7 cm.
2. The incircle of triangle ABC touches AB at D, BC at E and CA at F. AF = 3 cm, BD = 5 cm and CE = 4 cm. Find BC.
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Answer: B) 9 cm
BE = BD = 5 cm (tangents from B). BC = BE + EC = 5 + 4 = 9 cm. 8 cm uses AF + BD by mistake.
3. Tanu has a circle of radius 3 cm and a tangent PA = 4 cm. She writes OP = 3 + 4 = 7 cm. What is the correct OP?
- A5 cm, using OP² = OA² + PA²
- B12 cm, using 3 × 4
- C1 cm, using 4 − 3
- D7 cm, her answer is right
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Answer: A) 5 cm, using OP² = OA² + PA²
OA ⊥ PA, so triangle OAP is right-angled at A:
OP=9+16=5 cm. Adding the lengths treats O, A, P as if they were on one line.
4. A circle touches the sides AB, BC, CD, DA of quadrilateral ABCD at P, Q, R, S. AB = 6 cm, BC = 9 cm, CD = 8 cm and AP = 2 cm. Find DR.
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Answer: C) 3 cm
BP = 6 − 2 = 4 = BQ. CQ = 9 − 4 = 5 = CR. DR = 8 − 5 = 3 cm. (Check: AD = AS + SD = 2 + 3 = 5 and 6 + 8 = 9 + 5.)
5. A chord AB = 24 cm of the outer of two concentric circles touches the inner circle. The radii differ by 4 cm. Find the outer radius.
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Answer: B) 20 cm
R2−r2=122=144 and R − r = 4, so R + r = 144 ÷ 4 = 36. R = (36 + 4)/2 = 20 cm (inner radius 16 cm).