Points P(6, 0), Q(2, 8) and R(-2, 4) are vertices of △PQR. It is given that MN \ QR such that PM/MQ = 1/3. Using distance formula and ratio formula, show that MN/QR = 1/4.

M(5, 2), N(4, 1), MN = 2, QR = 42, so MN/QR = 14

MathsCBSE Class 10 · Basic · Board exam 2026, set 430/5/1 · 3 marks · Short answer

CBSE Class 10 Mathematics Basic 2026: question 27

Set in the CBSE Class 10 board exam of 2026, paper 430/5/1, section C, question 27, for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q27 · 3 marks · Short answerCoordinate Geometry

Points P(6, 0), Q(2, 8) and R(-2, 4) are vertices of $△$ PQR. It is given that MN $∥$ QR such that $\frac{P M}{M Q} = \frac{1}{3}$ . Using distance formula and ratio formula, show that $\frac{M N}{QR} = \frac{1}{4}$ .

Figure in the paper: Triangle PQR with P(6, 0) at the top, Q(2, 8) at bottom left and R(-2, 4) at bottom right. Point M lies on side PQ and point N lies on side PR, with MN drawn parallel to QR (arrows on MN and QR). See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2026 board paper, set 430/5/1.

Hints

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

Hint 1

Since MN ∥ QR, N divides PR in the same ratio as M divides PQ (Basic Proportionality Theorem).

Hint 2

Apply the section formula with ratio 1 : 3 starting from P, then use the distance formula for MN and QR.

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.

$M N ∥ QR$ , so by BPT $\frac{P M}{M Q} = \frac{P N}{N R} = \frac{1}{3}$ ; M and N divide PQ and PR in ratio $1 : 3$ .
½ mark
$M = (\frac{1 \times 2 + 3 \times 6}{1 + 3}, \frac{1 \times 8 + 3 \times 0}{1 + 3}) = (\frac{20}{4}, \frac{8}{4}) = (5, 2)$
½ mark
$N = (\frac{1 \times ( - 2 ) + 3 \times 6}{4}, \frac{1 \times 4 + 3 \times 0}{4}) = (\frac{16}{4}, \frac{4}{4}) = (4, 1)$
½ mark
$M N = (5 - 4)^{2} + (2 - 1)^{2} = 2$
½ mark
$QR = (2 + 2)^{2} + (8 - 4)^{2} = 32 = 42$
½ mark
$\frac{M N}{QR} = \frac{2}{4 2} = \frac{1}{4}$ . Hence shown.
½ mark

Answer: M(5, 2), N(4, 1), MN = $2$ , QR = $42$ , so $\frac{M N}{QR} = \frac{1}{4}$

Where the marks are

Uses BPT: $\frac{P N}{N R} = \frac{1}{3}$ (½ mark)
M = (5, 2) by section formula (½ mark)
N = (4, 1) by section formula (½ mark)
MN = $2$ (½ mark)
QR = $42$ (½ mark)
Concludes ratio $\frac{1}{4}$ (½ mark)

Revise the chapter

Chapter: Coordinate Geometry (notes, formulas, where marks are lost and every past question)
Topic: Section Formula (Internal Division)

Practise this chapterBoard-style questions from Coordinate Geometry, one at a time, marked against the answer key and marking points.Back to the full paper2026 board paper, set 430/5/1 (Basic): 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 Basic 2026: question 27

Hints

Step-by-step solution

Revise the chapter

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