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Mathematics · 2026 · 3 marks
CBSE 2026 · Region 2 · Set 1 · Q29
(a) Prove that the lengths of tangents drawn from an external point to a circle are equal.
(a)
Prove that the lengths of tangents drawn from an external point to a circle are equal.
OR
(b)
Two tangents TP and TQ are drawn to a circle with centre O from an external point T . Prove that $\displaystyle \angle \mathrm{PTQ}=2 \angle \mathrm{OPQ}$.
Marking-scheme solution
for correct Given, To Prove
for correct Proof
\(\displaystyle \mathrm{OP}=\mathrm{OQ} \Rightarrow \angle \mathrm{OPQ}=\angle \mathrm{OQP}\)
\(\displaystyle \angle \mathrm{OPQ}+\angle \mathrm{OQP}+\angle \mathrm{POQ}=180^{\circ}\)
\(\displaystyle \Rightarrow \angle \mathrm{POQ}=180^{\circ}-2 \angle \mathrm{OPQ}\)------(i)
Also, \(\displaystyle \angle \mathrm{OPT}+\angle \mathrm{OQT}+\angle \mathrm{POQ}+\angle \mathrm{PTQ}=360^{\circ}\)
\(\displaystyle \Rightarrow \angle \mathrm{POQ}=180^{\circ}-\angle \mathrm{PTQ}\)------(ii)
By (i) and (ii)
\(\displaystyle 180^{\circ}-2 \angle \mathrm{OPQ}=180^{\circ}-\angle \mathrm{PTQ}\)
\(\displaystyle \Rightarrow 2 \angle \mathrm{OPQ}=\angle \mathrm{PTQ}\)
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CBSE Class 10 Mathematics past-paper question from the 2026 board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.