AD and PM are medians of triangles ABC and PQR, respectively, where △ABC ~ △PQR. Prove that AB/PQ = AD/PM.

△ ABD ~ △ PQM (SAS), hence AB/PQ = AD/PM (proved)

MathsCBSE Class 10 · Basic · Board exam 2024, set 430/2/1 · 5 marks · Long answer

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

Set in the CBSE Class 10 board exam of 2024, paper 430/2/1, section D, question 34(b), for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q34(b) · 5 marks · Long answerTriangles

AD and PM are medians of triangles ABC and PQR, respectively, where $△$ ABC $\sim$ $△$ PQR. 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 2024 board paper, set 430/2/1.

Hints

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

Hint 1

From the similar triangles, write the ratio of AB to PQ and of BC to QR.

Hint 2

A median meets the opposite side at its mid-point, so replace BC and QR by twice BD and twice QM.

Hint 3

Now compare triangles ABD and PQM: two proportional sides and an included angle give which criterion?

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 the mid-point of BC) and $△ P QR$ with median PM (M the mid-point of QR). Given: $△ A B C \sim △ P QR$ . To prove: $\frac{A B}{P Q} = \frac{A D}{P M}$ .
1 mark
Since $△ A B C \sim △ P QR$ , corresponding sides are proportional: $\frac{A B}{P Q} = \frac{B C}{QR}$
1 mark
$B C = 2 B D$ and $QR = 2 QM$ [D and M are mid-points], so $\frac{A B}{P Q} = \frac{2 B D}{2 QM} = \frac{B D}{QM}$
1 mark
In $△ A B D$ and $△ P QM$ : $\frac{A B}{P Q} = \frac{B D}{QM}$ [proved above]
½ mark
$∠ B = ∠ Q$ [corresponding angles of similar triangles ABC and PQR]
½ mark
Therefore $△ A B D \sim △ P QM$ [SAS similarity].
½ mark
Hence corresponding sides are proportional: $\frac{A B}{P Q} = \frac{A D}{P M}$ .
½ mark

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

Where the marks are

Figure with given and to prove (1 mark)
$\frac{A B}{P Q} = \frac{B C}{QR}$ from the similar triangles (1 mark)
Uses the mid-points to get $\frac{A B}{P Q} = \frac{B D}{QM}$ (1 mark)
Ratio stated for triangles ABD and PQM (½ mark)
$∠ B = ∠ Q$ (½ mark)
$△ A B D \sim △ P QM$ by SAS (½ mark)
Concludes $\frac{A B}{P Q} = \frac{A D}{P M}$ (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 430/2/1 (Basic): every question, by section.

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

Hints

Step-by-step solution

Revise the chapter

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