A group of class X students goes to picnic during winter holidays. The position of three friends Aman, Kirti and Chahat are shown by the points P, Q and R. For Visually Impaired Candidates: The co-ordinates of P (2,5), Q (4,4) and R (8,3) are given. (i) Find the distance between P and R. (ii) Is Q, the midpoint of PR? Justify by finding midpoint of PR. (iii) (A) Find the point on x-axis which is equidistant from P and Q. (iii) (B) Let S be a point which divides the line joining PQ in ratio 2:3. Find the coordinates of S.

(i) 2√(10) units; (ii) No, midpoint of PR is (5,4) ≠ Q(4,4); (iii)(a) (34, 0), (iii)(b) (14/5, 23/5)

MathsCBSE Class 10 · Standard · Sample paper, session 2025-26 · 4 marks · Case-based

CBSE Class 10 Mathematics Standard sample paper 2025-26: question 37

From the official CBSE sample question paper for 2025-26, section E, question 37, for 4 marks. Try it first; the hints and our step-by-step solution are below.

Q37 · 4 marks · Case-basedCompetency-basedCoordinate Geometry

A group of class X students goes to picnic during winter holidays. The position of three friends Aman, Kirti and Chahat are shown by the points P, Q and R.

For Visually Impaired Candidates: The co-ordinates of P (2,5), Q (4,4) and R (8,3) are given.

(i)
Find the distance between P and R.
1 mark
(ii)
Is Q, the midpoint of PR? Justify by finding midpoint of PR.
1 mark
(iii)
(A) Find the point on x-axis which is equidistant from P and Q.
2 marks
OR
(iii)
(B) Let S be a point which divides the line joining PQ in ratio 2:3. Find the coordinates of S.
2 marks

Figure in the paper: Coordinate grid with x-axis marked 0 to 9 and y-axis 1 to 6. Points P(2,5), Q(4,4), R(8,3) are marked with dotted lines to the axes, and P, Q, R are joined by a line segment. See it in the official paper.

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2025-26.

Hints

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

Hint 1

Read the coordinates P(2,5), Q(4,4), R(8,3) from the graph.

Hint 2

Any point on the x-axis has the form $(x, 0)$ . Squaring both distances removes the square roots.

Hint 3

Section formula: $(\frac{m x _{2} + n x _{1}}{m + n}, \frac{m y _{2} + n y _{1}}{m + n})$ .

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.

(i) With $P (2, 5)$ and $R (8, 3)$ , by the distance formula:
$P R = (8 - 2)^{2} + (3 - 5)^{2} = 36 + 4 = 40 = 210$ units
1 mark
(ii) From the figure, $Q = (4, 4)$ .
Midpoint of $P R = (\frac{2 + 8}{2}, \frac{5 + 3}{2}) = (5, 4)$
½ mark
(ii) Since $(5, 4) \neq = (4, 4)$ , Q is not the midpoint of PR.
½ mark
(iii)(a) Let the point on the x-axis be $A (x, 0)$ . Set $A P = A Q$ , so $A P^{2} = A Q^{2}$ :
$(x - 2)^{2} + 25 = (x - 4)^{2} + 16$
1 mark
(iii)(a) $x^{2} - 4 x + 29 = x^{2} - 8 x + 32$
$\Rightarrow 4 x = 3$
$\Rightarrow x = \frac{3}{4}$
The required point is $(\frac{3}{4}, 0)$ .
1 mark
(iii)(b) S divides $P (2, 5)$ and $Q (4, 4)$ in the ratio $m : n = 2 : 3$ . By the section formula:
$S = (\frac{2 \times 4 + 3 \times 2}{2 + 3}, \frac{2 \times 4 + 3 \times 5}{2 + 3})$
1 mark
(iii)(b) $S = (\frac{14}{5}, \frac{23}{5})$
1 mark

Answer: (i) $210$ units; (ii) No, midpoint of PR is (5,4) ≠ Q(4,4); (iii)(a) $(\frac{3}{4}, 0)$ , (iii)(b) $(\frac{14}{5}, \frac{23}{5})$

Where the marks are

(i) $P R = 210$ (1 mark)
(ii) Midpoint (5,4) found (½ mark); concludes Q is not the midpoint (½ mark)
(iii)(a) Equates squared distances from $(x, 0)$ (1 mark); gets $(\frac{3}{4}, 0)$ (1 mark)
(iii)(b) Correct section formula substitution (1 mark); $S = (\frac{14}{5}, \frac{23}{5})$ (1 mark)

Revise the chapter

Chapter: Coordinate Geometry (notes, formulas, where marks are lost and every past question)
Topic: Applications of the Distance Formula

Practise this chapterBoard-style questions from Coordinate Geometry, one at a time, marked against the answer key and marking points.Back to the full paperSample paper 2025-26 (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 sample paper 2025-26: question 37

Hints

Step-by-step solution

Revise the chapter

More questions from Coordinate Geometry