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Mathematics · 2026 · 3 marks
CBSE 2026 · Region 1 · Set 1 · Q29 (B)
In the given figure, if a circle touches the side QR of PQR at S and extended sides PQ and PR at M and N respectively, then prove that: PM =1/2( PQ +…
In the given figure, if a circle touches the side QR of $\displaystyle \triangle \mathrm{PQR}$ at S and extended sides PQ and PR at M and N respectively, then prove that : $\displaystyle \mathrm{PM}=\frac{1}{2}(\mathrm{PQ}+\mathrm{QR}+\mathrm{PR})$
Marking-scheme solution
\[\left.\begin{array}{l}
\mathrm{PM}=\mathrm{PN} \\
\mathrm{QS}=\mathrm{QM} \\
\mathrm{RS}=\mathrm{RN}
\end{array}\right\}
\]
\[\begin{aligned}
\mathrm{PM}+\mathrm{PN} & =\mathrm{PQ}+\mathrm{QM}+\mathrm{PR}+\mathrm{RN} \\
2 \mathrm{PM} & =\mathrm{PQ}+\mathrm{QS}+\mathrm{PR}+\mathrm{RS} \\
& =\mathrm{PQ}+\mathrm{QS}+\mathrm{RS}+\mathrm{PR}
\end{aligned}
\]
\[=\mathrm{PQ}+\mathrm{QR}+\mathrm{PR}
\]
\[\therefore P M=\frac{1}{2}(P Q+Q R+P R)
\]More from Circles
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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.