✓ Board-verified
Mathematics · 2025 · 2 marks
CBSE 2025 · Region 2 · Set 3 · Q25
(a) In the given figure, PS/SQ=PT/TR and ∠ PST =∠ PRQ. Prove that Δ PQR is an isosceles triangle.
In the given figure, $\displaystyle \frac{\mathrm{PS}}{\mathrm{SQ}}=\frac{\mathrm{PT}}{\mathrm{TR}}$ and $\displaystyle \angle \mathrm{PST}=\angle \mathrm{PRQ}$. Prove that $\displaystyle \Delta \mathrm{PQR}$ is an isosceles triangle.
In the given figure, $\displaystyle \Delta \mathrm{ABE} \cong \Delta \mathrm{ACD}$. Prove that $\displaystyle \Delta \mathrm{ADE} \sim \Delta \mathrm{ABC}$.
Marking-scheme solution
Given \(\displaystyle \frac{\mathrm{PS}}{\mathrm{SQ}}=\frac{\mathrm{PT}}{\mathrm{TR}}\)
\[\begin{aligned}
& \Rightarrow \mathrm{ST} \| \mathrm{QR} \\
& \therefore \angle \mathrm{PST}=\angle \mathrm{PQR}
\end{aligned}
\]
and given, \(\displaystyle \angle \mathrm{PST}=\angle \mathrm{PRQ}\)
So, \(\displaystyle \angle \mathrm{PQR}=\angle \mathrm{PRQ}\)
\(\displaystyle \therefore \triangle \mathrm{PQR}\) is an isosceles triangle.
Given \(\displaystyle \Delta \mathrm{ABE} \cong \Delta \mathrm{ACD}\)
\[\therefore \mathrm{AE}=\mathrm{AD} \text { or } \mathrm{AD}=\mathrm{AE} \quad \text {---- ① }
\]
and \(\displaystyle \mathrm{AB}=\mathrm{AC} \quad\) ---- ②
Dividing ① by ②, we have
\[\frac{A D}{A B}=\frac{A E}{A C}
\]
and \(\displaystyle \angle \mathrm{DAE}=\angle \mathrm{BAC}\)
\[\therefore \triangle \mathrm{ADE} \sim \triangle \mathrm{ABC}
\]
More from Triangles
- Sides AB and AC and median AM of a ABC are proportional to sides DE and DF and Median DN of another DEF. Show that ABC ∼ DEF. ABCD is a…2023
- In the given figure, DE ∥ BC. If AD=2 units, DB=AE=3 units and EC=x units, then the value of x is:2023
- Which of the following statements is incorrect?2025
- E is a point on the side AD produced of a parallelogram ABCD and BE intersects CD at F. Show that Δ ABE ∼ Δ CFB. OR Sides AB, BC and the median AD of…2024
- If PQR ∼ ABC; PQ=6 cm, AB=8 cm and the perimeter of ABC is 36 cm, then the perimeter of PQR is2023
- In the given figure, ABCD is a quadrilateral. Diagonal BD bisects ∠ B and ∠ D both. Prove that:2024
- E and F are points on the sides AB and AC respectively of a ABC such that AE/EB=AF/FC=1/2. Which of the following relation is true?2025
- If Δ ABC ∼ Δ PQR with ∠ A=32^° and ∠ R=65^°, then the measure of ∠ B is:2023
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.