Q32. Prove that the lengths of tangents drawn from an external point to a circle are equal.

In right triangles OQP and ORP: OQ = OR (radii), OP is common and ∠OQP = ∠ORP = 90°, so △OQP ≅ △ORP by RHS; hence PQ = PR by CPCT, i.e. the tangents drawn from an external point to a circle are equal in length.

CBSE Class 10 Maths (Basic) 2024 — Paper 430(B)/S

80 marks · 3 hours · 38 questions (44 with the OR choices)

1 of its 44 questions has been set in other papers too (each one says where); the other 43 are only in this paper.

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Also in 2026 · 30/2/1 (Q29(a)), 2026 · 430/1/1 (Q30), 2023 · 430/4/1 (Q31(a)), 2020 · 430(B) (Q37(a)), 2018 · 30/1/1 (Q18), 2018 · 30/2/1 (Q15), 2018 · 30/3/1 (Q16), 2017 · 30/1/1 (Q22), 2017 · 30(B) (Q24), 2017 · 530/1 (Q24), 2016 · 30/1/1 (Q22), 2016 · 30/2/1 (Q22), 2016 · 30(B) (Q24), 2016 · 530/1 (Q24), 2015 · 30(B) (Q24)

Prove that the lengths of tangents drawn from an external point to a circle are equal.

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In right triangles $O QP$ and $O R P$ : $O Q = O R$ (radii), $O P$ is common and $∠ O QP = ∠ O R P = 9 0^{\circ}$ , so $△ O QP ≅ △ O R P$ by RHS; hence $P Q = P R$ by CPCT, i.e. the tangents drawn from an external point to a circle are equal in length.

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