MathsCBSE Class 10 · Standard · Board exam 2023, set 30/4/1 · 5 marks · Long answer

CBSE Class 10 Mathematics Standard 2023: question 33(B)

Set in the CBSE Class 10 board exam of 2023, 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

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

Source: official question paper from cbse.gov.in (PDF). See it in the full 2023 board paper, set 30/4/1.

Hints

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

Hint 1

Try to find two smaller triangles, one in each figure, that contain the medians AD and PM.

Hint 2

Write the ratio $\frac{A B}{P Q} = \frac{B C}{QR}$ , then replace BC and QR by twice BD and twice QM.

Hint 3

Use $∠ B = ∠ Q$ with that ratio to apply SAS similarity to $△ A B D$ and $△ P QM$ .

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: Draw $△ A B C$ with median AD (D is the midpoint of BC). Draw $△ P QR$ with median PM (M is the midpoint of QR).
Given: $△ A B C \sim △ P QR$ . To prove: $\frac{A B}{P Q} = \frac{A D}{P M}$ .
1 mark
Since $△ A B C \sim △ P QR$ , their corresponding sides are proportional:
$\frac{A B}{P Q} = \frac{B C}{QR}$
D and M are midpoints, so $B C = 2 B D$ and $QR = 2 QM$ .
$\Rightarrow \frac{A B}{P Q} = \frac{2 B D}{2 QM} = \frac{B D}{QM}$ ...(1)
1½ marks
Also $∠ B = ∠ Q$ , because corresponding angles of similar triangles are equal.
In $△ A B D$ and $△ P QM$ :
- $\frac{A B}{P Q} = \frac{B D}{QM}$ [from (1)]
- $∠ B = ∠ Q$
$\Rightarrow △ A B D \sim △ P QM$ [SAS similarity criterion]
1½ marks
Corresponding sides of similar triangles are proportional, so
$\frac{A B}{P Q} = \frac{A D}{P M}$ . Hence proved.
1 mark

Answer: $△ A B D \sim △ P QM$ (SAS), hence $\frac{A B}{P Q} = \frac{A D}{P M}$ .

Where the marks are

Correct figure showing both medians (1 mark)
Uses similarity to get $\frac{A B}{P Q} = \frac{B C}{QR} = \frac{B D}{QM}$ with the halves of the sides (1½ marks)
Uses $∠ B = ∠ Q$ and proves $△ A B D \sim △ P QM$ by SAS (1½ marks)
Concludes $\frac{A B}{P Q} = \frac{A D}{P M}$ (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 paper2023 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 2023: question 33(B)

Hints

Step-by-step solution

Revise the chapter

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