MathsCBSE Class 10 · Standard · Board exam 2024, set 30/4/1 · 5 marks · Long answer

CBSE Class 10 Mathematics Standard 2024: question 33(b)

Set in the CBSE Class 10 board exam of 2024, paper 30/4/1, section D, question 33(b), for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q33(b) · 5 marks · Long answerTriangles

Sides AB, BC and the median AD of $△$ ABC are respectively proportional to sides PQ, QR and the median PM of another $△$ PQR. Prove that $△$ ABC $\sim$ $△$ PQR.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2024 board paper, set 30/4/1.

Hints

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

Hint 1

A median bisects the side, so BC = 2BD and QR = 2QM.

Hint 2

First prove the smaller triangles ABD and PQM similar, then use the equal angle in the big triangles.

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.

Figure: $△ A B C$ with median AD (D mid-point of BC) and $△ P QR$ with median PM (M mid-point of QR). Given $\frac{A B}{P Q} = \frac{B C}{QR} = \frac{A D}{P M}$ .
1 mark
Since $B C = 2 B D$ and $QR = 2 QM$ : $\frac{A B}{P Q} = \frac{2 B D}{2 QM} = \frac{A D}{P M} \Rightarrow \frac{A B}{P Q} = \frac{B D}{QM} = \frac{A D}{P M}$ ...(i)
1 mark
From (i), $△ A B D \sim △ P QM$ [SSS similarity criterion].
1 mark
Hence $∠ B = ∠ Q$ [corresponding angles of similar triangles] ...(ii)
1 mark
In $△ A B C$ and $△ P QR$ : $\frac{A B}{P Q} = \frac{B C}{QR}$ (given) and $∠ B = ∠ Q$ from (ii). Therefore $△ A B C \sim △ P QR$ [SAS similarity criterion].
1 mark

Answer: △ABC ∼ △PQR (via △ABD ∼ △PQM by SSS, then SAS)

Where the marks are

Correct figure with medians (1 mark)
Uses BC = 2BD, QR = 2QM to get AB/PQ = BD/QM = AD/PM (1 mark)
△ABD ∼ △PQM by SSS (1 mark)
∠B = ∠Q (1 mark)
△ABC ∼ △PQR by SAS (1 mark)

Revise the chapter

Chapter: Triangles (notes, formulas, where marks are lost and every past question)
Topic: Problems on Similar Triangles

Practise this chapterBoard-style questions from Triangles, one at a time, marked against the answer key and marking points.Back to the full paper2024 board paper, set 30/4/1 (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 2024: question 33(b)

Hints

Step-by-step solution

Revise the chapter

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