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Mathematics · 2025 · 5 marks
CBSE 2025 · Region 6 · Set 1 · Q33
(a) If a line drawn parallel to one side of triangle intersecting the other two sides in distinct points divides the two sides in the same ratio,…
If a line drawn parallel to one side of triangle intersecting the other two sides in distinct points divides the two sides in the same ratio, then it is parallel to third side. State and prove the converse of the above statement.
In the adjoining figure, $\displaystyle \triangle \mathrm{CAB}$ is a right triangle, right angled at A and $\displaystyle \mathrm{AD} \perp \mathrm{BC}$. Prove that $\displaystyle \Delta \mathrm{ADB} \sim \Delta \mathrm{CDA}$. Further, if $\displaystyle \mathrm{BC}=10 \mathrm{~cm}$ and $\displaystyle \mathrm{CD}=2 \mathrm{~cm}$, find the length of AD.
Marking-scheme solution
(b)
\[\begin{aligned}
& \Delta \mathrm{ABC} \sim \Delta \mathrm{DAC} \quad ---(1) \\
& \text{Similarly, } \Delta \mathrm{ABC} \sim \Delta \mathrm{DBA} \quad ---(2)
\end{aligned}
\]
From equations ($\displaystyle 1$) and ($\displaystyle 2$)
\[\begin{aligned}
& \Delta \mathrm{DAC} \sim \Delta \mathrm{DBA} \text{ or } \Delta \mathrm{ADB} \sim \Delta \mathrm{CDA} \\
& \frac{\mathrm{AD}}{\mathrm{CD}}=\frac{\mathrm{BD}}{\mathrm{AD}} \\
& \mathrm{AD}^{2}=\mathrm{BD} \times \mathrm{CD} \\
& \quad\ =8 \times 2 \\
& \therefore \mathrm{AD}=4 \mathrm{~cm}.
\end{aligned}
\]
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CBSE Class 10 Mathematics past-paper question from the 2025 board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.