MathsCBSE Class 10 · Standard · Sample paper, session 2024-25 · 3 marks · Short answer

CBSE Class 10 Mathematics Standard sample paper 2024-25: question 26(B)

From the official CBSE sample question paper for 2024-25, section C, question 26(B), for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q26(B) · 3 marks · Short answerTriangles

In $△ A B C$ , $P$ and $Q$ are points on $A B$ and $A C$ respectively such that $P Q$ is parallel to $B C$ . Prove that the median $A D$ drawn from $A$ on $B C$ bisects $P Q$ .

Figure in the paper: Triangle with apex A at top, B bottom-left, C bottom-right, D on BC (bottom). P lies on AB and Q lies on AC, with segment PQ drawn horizontally. R is the point where median AD meets PQ. The median AD is drawn from A to D. See it in the official paper.

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2024-25.

Hints

Open one hint at a time and try again before the next.

Hint 1

Look for two pairs of similar triangles that have vertex A and lie along AD.

Hint 2

Show $△ A P R \sim △ A B D$ and $△ A QR \sim △ A C D$ . Both give the same ratio $\frac{A R}{A D}$ ; then use $B D = D C$ .

Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

Given: $P Q ∥ B C$ , $A D$ is a median ( $B D = D C$ ), and $A D$ meets $P Q$ at $R$ . To prove: $P R = R Q$ .
In $△ A P R$ and $△ A B D$ : $∠ P A R = ∠ B A D$ (common), and $∠ A P R = ∠ A B D$ [corresponding angles, $P Q ∥ B C$ ].
So $△ A P R \sim △ A B D$ [AA], which gives $\frac{A P}{A B} = \frac{P R}{B D} = \frac{A R}{A D}$ …(i)
1 mark
Similarly, in $△ A QR$ and $△ A C D$ : $∠ Q A R = ∠ C A D$ (common), and $∠ A QR = ∠ A C D$ [corresponding angles].
So $△ A QR \sim △ A C D$ [AA], which gives $\frac{A Q}{A C} = \frac{R Q}{D C} = \frac{A R}{A D}$ …(ii)
1 mark
From (i) and (ii), both ratios equal $\frac{A R}{A D}$ , so $\frac{P R}{B D} = \frac{R Q}{D C}$ .
Since $B D = D C$ [ $A D$ is a median], $P R = R Q$ . Hence the median $A D$ bisects $P Q$ .
1 mark

Answer: $P R = R Q$ , so the median $A D$ bisects $P Q$ .

Where the marks are

Proves $△ A P R \sim △ A B D$ and writes the ratio of sides (1 mark)
Proves $△ A QR \sim △ A C D$ and writes the ratio of sides (1 mark)
Equates the ratios and uses $B D = D C$ to conclude $P R = R Q$ (1 mark)

Revise the chapter

Chapter: Triangles (notes, formulas, where marks are lost and every past question)
Topic: Basic Proportionality Theorem (Thales Theorem)

Practise this chapterBoard-style questions from Triangles, one at a time, marked against the answer key and marking points.Back to the full paperSample paper 2024-25 (Standard): every question, by section.

The question is transcribed from the official paper; figures are described in words. The solution, hints and chapter tag are ours. Spotted an error? Write tosupport@questionbuddy.in.

CBSE Class 10 Mathematics Standard sample paper 2024-25: question 26(B)

Hints

Step-by-step solution

Revise the chapter

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