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Mathematics · 2023 · 3 marks
CBSE 2023 · Region 4 · Set 3 · Q27
From an external point, two tangents are drawn to a circle. Prove that the line joining the external point to the centre of the circle bisects the…
From an external point, two tangents are drawn to a circle. Prove that the line joining the external point to the centre of the circle bisects the angle between the two tangents.
Marking-scheme solution
Given : PA and PB are tangents drawn from an external point P to the circle with centre O.
To prove: \(\displaystyle \angle \mathrm{OPA}=\angle \mathrm{OPB}\)
Construction: Join OA, OB
Proof: In \(\displaystyle \Delta \mathrm{OPA}\) and \(\displaystyle \Delta \mathrm{OPB}\)
\[\begin{aligned}
& \mathrm{OP}=\mathrm{OP} \text { (common) } \\
& \mathrm{OA}=\mathrm{OA} \text { (radii) } \\
& \angle \mathrm{OAP}=\angle \mathrm{OBP} \text { (each } 90^{\circ}, \text { radius ⟂ tangents) } \\
& \therefore \Delta \mathrm{OPA} \cong \triangle \mathrm{OPB} \text { (RHS) } \\
& \Rightarrow \angle \mathrm{OPA}=\angle \mathrm{OPB} \text { (CPCT) }
\end{aligned}
\]
To prove: \(\displaystyle \angle \mathrm{OPA}=\angle \mathrm{OPB}\)
Construction: Join OA, OB
Proof: In \(\displaystyle \Delta \mathrm{OPA}\) and \(\displaystyle \Delta \mathrm{OPB}\)
\[\begin{aligned}
& \mathrm{OP}=\mathrm{OP} \text { (common) } \\
& \mathrm{OA}=\mathrm{OA} \text { (radii) } \\
& \angle \mathrm{OAP}=\angle \mathrm{OBP} \text { (each } 90^{\circ}, \text { radius ⟂ tangents) } \\
& \therefore \Delta \mathrm{OPA} \cong \triangle \mathrm{OPB} \text { (RHS) } \\
& \Rightarrow \angle \mathrm{OPA}=\angle \mathrm{OPB} \text { (CPCT) }
\end{aligned}
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CBSE Class 10 Mathematics past-paper question from the 2023 board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.