CBSE 2025 · Region 6 · Set 1 · Q33 · 5 marks
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.
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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