MathsCBSE Class 10 · Standard · Board exam 2024, set 30/5/1 · 4 marks · Case-based

CBSE Class 10 Mathematics Standard 2024: question 37

Set in the CBSE Class 10 board exam of 2024, paper 30/5/1, 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

Case Study – 2

A garden is in the shape of a square. The gardener grew saplings of Ashoka tree on the boundary of the garden at the distance of 1 m from each other. He wants to decorate the garden with rose plants. He chose a triangular region inside the garden to grow rose plants. In the above situation, the gardener took help from the students of class 10. They made a chart for it which looks like the given figure.

Based on the above, answer the following questions :

(i)
If A is taken as origin, what are the coordinates of the vertices of $△$ PQR ?
1 mark
(ii)(a)
Find distances PQ and QR.
2 marks
OR
(ii)(b)
Find the coordinates of the point which divides the line segment joining points P and R in the ratio 2 : 1 internally.
2 marks
(iii)
Find out if $△$ PQR is an isosceles triangle.
1 mark

Figure in the paper: A square garden ABCD with A at bottom-left, B at top-left, C at top-right and D at bottom-right. Ashoka saplings are along the boundary at 1 m intervals. A grid of 1 m squares is drawn inside. A shaded triangle PQR is inside the garden: P is above and to the right of Q, R is to the right of P at the same level as the grid line near P's lower height, with Q at the bottom. Labels P, Q and R mark the vertices. See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2024 board paper, set 30/5/1.

Hints

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

Hint 1

Count grid squares from A: across gives x, up gives y.

Hint 2

Use $(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}$ for distances and $(\frac{m x _{2} + n x _{1}}{m + n}, \frac{m y _{2} + n y _{1}}{m + n})$ for the dividing point.

Hint 3

For (iii), find all three sides and check whether any two are equal.

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) Taking A as origin, AD along the x-axis and AB along the y-axis (each grid square = 1 m), the vertices are $P (4, 6)$ , $Q (3, 2)$ , $R (6, 5)$ .
1 mark
(ii)(a) By the distance formula, $P Q = (4 - 3)^{2} + (6 - 2)^{2} = 1 + 16 = 17$ m.
1 mark
(ii)(a) $QR = (6 - 3)^{2} + (5 - 2)^{2} = 9 + 9 = 18 = 32$ m.
1 mark
(ii)(b) By the section formula with $m : n = 2 : 1$ , $P (4, 6)$ , $R (6, 5)$ , the point is $(\frac{2 \times 6 + 1 \times 4}{3}, \frac{2 \times 5 + 1 \times 6}{3})$ .
1 mark
(ii)(b) $= (\frac{16}{3}, \frac{16}{3})$ .
1 mark
(iii) $P R = (6 - 4)^{2} + (5 - 6)^{2} = 4 + 1 = 5$ m, and from (ii)(a) $P Q = 17$ m, $QR = 18$ m.
½ mark
(iii) Since $P Q \neq = QR \neq = P R$ , no two sides are equal, so $△ P QR$ is not an isosceles triangle.
½ mark

Answer: (i) P(4, 6), Q(3, 2), R(6, 5); (ii)(a) PQ = √17 m, QR = √18 = 3√2 m; (ii)(b) (16/3, 16/3); (iii) PR = √5, all sides unequal, so not isosceles

Where the marks are

(i) Writes the correct coordinates of P, Q and R (1 mark)
(ii)(a) $P Q = 17$ (1 mark)
(ii)(a) $QR = 18$ (1 mark)
(ii)(b) Correct substitution in the section formula (1 mark)
(ii)(b) Gets $(16/3, 16/3)$ (1 mark)
(iii) Finds $P R = 5$ (½ mark)
(iii) Concludes the sides are unequal, so the triangle is not isosceles (½ mark)

Revise the chapter

Chapter: Coordinate Geometry (notes, formulas, where marks are lost and every past question)
Topic: 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 paper2024 board paper, set 30/5/1 (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 2024: question 37

Hints

Step-by-step solution

Revise the chapter

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