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Mathematics · 2024 · 5 marks
CBSE 2024 · Region 1 · Set 1 · Q34
(A) 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…
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.
In the given figure PA, QB and RC are each perpendicular to AC. If $\displaystyle \mathrm{AP}=x, \mathrm{BQ}=y$ and $\displaystyle \mathrm{CR}=z$, then prove that $\displaystyle \frac{1}{x}+\frac{1}{z}=\frac{1}{y}$
Marking-scheme solution
Correct proof
\(\displaystyle \Delta \mathrm{PAC} \sim \Delta \mathrm{QBC}\)
\[\therefore \frac{x}{y}=\frac{A C}{B C} \text { or } \frac{y}{x}=\frac{B C}{A C}
\]
\[\begin{aligned}
& \Delta \mathrm{RCA} \sim \Delta \mathrm{QBA} \\
& \therefore \frac{z}{y}=\frac{A C}{A B} \text { or } \frac{y}{z}=\frac{A B}{A C}
\end{aligned}
\]
Adding (i) and (ii)
\[\begin{aligned}
& \frac{y}{x}+\frac{y}{z}=\frac{B C+A B}{A C} \\
& \Rightarrow \frac{1}{x}+\frac{1}{z}=\frac{1}{y}
\end{aligned}
\]
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CBSE Class 10 Mathematics past-paper question from the 2024 board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.