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Mathematics · 2024 · 5 marks
CBSE 2024 · Region 2 · Set 1 · Q33
If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then prove that the other two sides are divided in the same ratio.Sides AB and AC and median AD to $\displaystyle \triangle \mathrm{ABC}$ are respectively proportional to sides PQ and PR and median PM of another triangle PQR . Show that $\displaystyle \triangle \mathrm{ABC} \sim \triangle \mathrm{PQR}$.
If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then prove that the other two sides are divided in the same ratio.
Sides AB and AC and median AD to $\displaystyle \triangle \mathrm{ABC}$ are respectively proportional to sides PQ and PR and median PM of another triangle PQR . Show that $\displaystyle \triangle \mathrm{ABC} \sim \triangle \mathrm{PQR}$.
Marking-scheme solution
Correct Given, to prove, figure, construction Correct proof
Produce AD to E such that \(\displaystyle \mathrm{AD}=\mathrm{DE}\) and join EC Produce PM to N such that PM = MN and join NR \(\displaystyle \Delta \mathrm{ADB} \cong \triangle \mathrm{EDC} \therefore \mathrm{AB}=\mathrm{EC}\)
Similarly, \(\displaystyle \mathrm{PQ}=\mathrm{NR}\) Since, \(\displaystyle \frac{\mathrm{AB}}{\mathrm{PQ}}=\frac{\mathrm{AC}}{\mathrm{PR}}=\frac{\mathrm{AD}}{\mathrm{PM}} \Rightarrow \frac{\mathrm{EC}}{\mathrm{NR}}=\frac{\mathrm{AC}}{\mathrm{PR}}=\frac{\dfrac{\mathrm{AE}}{2}}{\dfrac{\mathrm{PN}}{2}} \therefore \triangle \mathrm{AEC} \sim \triangle \mathrm{PNR} \Rightarrow \angle 1=\angle 2\)
Similarly, \(\displaystyle \angle 3=\angle 4\) Hence \(\displaystyle \angle 1+\angle 3=\angle 2+\angle 4\) or \(\displaystyle \angle \mathrm{A}=\angle \mathrm{P}\) Also, \(\displaystyle \frac{\mathrm{AB}}{\mathrm{PQ}}=\frac{\mathrm{AC}}{\mathrm{PR}} \therefore \triangle \mathrm{ABC} \sim \triangle \mathrm{PQR}\)
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CBSE Class 10 Mathematics past-paper question from the 2024board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.