CBSE 2025 · Region 1 · Set 1 · Q33 · 5 marks
The diagonal BD of a parallelogram ABCD intersects the line segment AE at the point F, where E is any point on the side BC. Prove that $\displaystyle \mathrm{DF} \times \mathrm{EF}=\mathrm{FB} \times \mathrm{FA}$.In $\displaystyle \triangle \mathrm{ABC}$, if $\displaystyle \mathrm{AD} \perp \mathrm{BC}$ and $\displaystyle \mathrm{AD}^{2}=\mathrm{BD} \times \mathrm{DC}$, then prove that $\displaystyle \angle \mathrm{BAC}=90^{\circ}$.
The diagonal BD of a parallelogram ABCD intersects the line segment AE at the point F, where E is any point on the side BC. Prove that $\displaystyle \mathrm{DF} \times \mathrm{EF}=\mathrm{FB} \times \mathrm{FA}$.
In $\displaystyle \triangle \mathrm{ABC}$, if $\displaystyle \mathrm{AD} \perp \mathrm{BC}$ and $\displaystyle \mathrm{AD}^{2}=\mathrm{BD} \times \mathrm{DC}$, then prove that $\displaystyle \angle \mathrm{BAC}=90^{\circ}$.
Marking-scheme solution
In \(\displaystyle \Delta \mathrm{ADF}\) and \(\displaystyle \Delta \mathrm{EBF}\),
\[\begin{aligned}
& \angle \mathrm{DFA}=\angle \mathrm{EFB} \\
& \angle \mathrm{ADF}=\angle \mathrm{FBE} \\
& \therefore \triangle \mathrm{ADF} \sim \triangle \mathrm{EBF} \\
& \therefore \frac{\mathrm{DF}}{\mathrm{FB}}=\frac{\mathrm{FA}}{\mathrm{EF}} \\
& \Rightarrow \mathrm{DF} \times \mathrm{EF}=\mathrm{FB} \times \mathrm{FA}
\end{aligned}
\]
\[\begin{aligned}
& \mathrm{AD}^{2}=\mathrm{BD} \times \mathrm{DC} \\
& \frac{\mathrm{AD}}{\mathrm{DC}}=\frac{\mathrm{BD}}{\mathrm{AD}}
\end{aligned}
\]
Also, \(\displaystyle \angle \mathrm{ADB}=\angle \mathrm{ADC}\)
\[\begin{aligned}
& \therefore \Delta \mathrm{DBA} \sim \Delta \mathrm{DAC} \\
& \angle \mathrm{DBA}=\angle \mathrm{DAC} \\
& \angle \mathrm{BAD}=\angle \mathrm{DCA}
\end{aligned}
\]
Adding both
\[\begin{aligned}
& \angle \mathrm{DBA}+\angle \mathrm{DCA}=\angle \mathrm{DAC}+\angle \mathrm{BAD} \\
& \therefore \angle \mathrm{BAC}=90^{\circ}
\end{aligned}
\]
TrianglesApplications of SimilarityAnalyselong_answerhard
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CBSE Class 10 Mathematics past-paper question from the 2025board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.