MathsCBSE Class 10 · Basic · Board exam 2024, set 430/4/1 · 5 marks · Long answer

CBSE Class 10 Mathematics Basic 2024: question 34(a)

Set in the CBSE Class 10 board exam of 2024, paper 430/4/1, section D, question 34(a), for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q34(a) · 5 marks · Long answerTriangles

If BD and QM are medians of triangles ABC and PQR, respectively, where $△ A B C \sim △ P QR$ , prove that $\frac{A B}{P Q} = \frac{B D}{QM}$ .

Source: official question paper from cbse.gov.in (PDF). See it in the full 2024 board paper, set 430/4/1.

Hints

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

Hint 1

The medians split AC and PR in half. Write AB/PQ in terms of AD and PM.

Hint 2

Show △ABD ~ △PQM using the SAS similarity criterion with ∠A = ∠P.

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 BD to side AC (D is the midpoint of AC), and $△ P QR$ with median QM to side PR (M is the midpoint of PR). Given: $△ A B C \sim △ P QR$ . To prove: $\frac{A B}{P Q} = \frac{B D}{QM}$ .
1 mark
Since $△ A B C \sim △ P QR$ , corresponding sides are proportional: $\frac{A B}{P Q} = \frac{A C}{P R}$ .
1 mark
As D and M are midpoints, $A C = 2 A D$ and $P R = 2 P M$ . So $\frac{A B}{P Q} = \frac{2 A D}{2 P M} = \frac{A D}{P M}$ .
1 mark
In $△ A B D$ and $△ P QM$ : $\frac{A B}{P Q} = \frac{A D}{P M}$ (proved above) and $∠ A = ∠ P$ [corresponding angles of similar triangles]. So $△ A B D \sim △ P QM$ [SAS similarity].
1½ marks
Corresponding sides of similar triangles are proportional, so $\frac{A B}{P Q} = \frac{B D}{QM}$ . Hence proved.
½ mark

Answer: △ABD ~ △PQM (SAS), hence AB/PQ = BD/QM.

Where the marks are

Correct figure with given and to prove (1 mark)
AB/PQ = AC/PR from the similarity (1 mark)
Uses the midpoints to get AB/PQ = AD/PM (1 mark)
Proves △ABD ~ △PQM by SAS using ∠A = ∠P (1½ marks)
Concludes AB/PQ = BD/QM (½ mark)

Revise the chapter

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

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 430/4/1 (Basic): every question, by section.

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CBSE Class 10 Mathematics Basic 2024: question 34(a)

Hints

Step-by-step solution

Revise the chapter

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