Ryan, from a very young age, was fascinated by the twinkling of stars and the vastness of space. He always dreamt of becoming an astronaut one day. So he started to sketch his own rocket designs on the graph sheet. One such design is given below : Based on the above, answer the following questions : (i) Find the mid-point of the segment joining F and G. (ii) (a) What is the distance between the points A and C ? (ii) (b) Find the coordinates of the point which divides the line segment joining the points A and B in the ratio 1 : 3 internally. (iii) What are the coordinates of the point D ?

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

CBSE Class 10 Mathematics Standard 2024: question 36

Set in the CBSE Class 10 board exam of 2024, paper 30/4/1, section E, question 36, for 4 marks. Try it first; the hints and our step-by-step solution are below.

Q36 · 4 marks · Case-basedCompetency-basedCoordinate Geometry

Ryan, from a very young age, was fascinated by the twinkling of stars and the vastness of space. He always dreamt of becoming an astronaut one day. So he started to sketch his own rocket designs on the graph sheet. One such design is given below :

Based on the above, answer the following questions :

(i)
Find the mid-point of the segment joining F and G.
1 mark
(ii)
(a) What is the distance between the points A and C ?
2 marks
OR
(ii)
(b) Find the coordinates of the point which divides the line segment joining the points A and B in the ratio $1 : 3$ internally.
2 marks
(iii)
What are the coordinates of the point D ?
1 mark

Figure in the paper: A rocket design drawn on a graph sheet with X and Y axes (x from -6 to 6, y from -5 to 5). Labelled vertices: A(3, 4), B(3, 2), C(-1, -2), D(-2, -5), E(-5, 0), F(-3, 0), G(1, 4). The rocket outline runs A to B on the right, B to C diagonally, C down to a notch near (-1, -4), up to D at the bottom, and then the left side with zigzag fins; segment EF and FG form the upper-left edge with G at (1, 4). See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2024 board paper, set 30/4/1.

Hints

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

Hint 1

Read the coordinates of each labelled point off the graph first, for example F(−3, 0) and G(1, 4).

Hint 2

Use the mid-point formula, the distance formula $(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}$ and the section formula $(\frac{m x _{2} + n x _{1}}{m + n}, \frac{m y _{2} + n y _{1}}{m + n})$ .

Hint 3

In the 1 : 3 division starting from A, the coordinates of B get weight 1 and those of A get weight 3.

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) From the graph, F(−3, 0) and G(1, 4). Mid-point $= (\frac{- 3 + 1}{2}, \frac{0 + 4}{2}) = (- 1, 2)$
1 mark
(ii)(a) From the graph, A(3, 4) and C(−1, −2). $A C = (- 1 - 3)^{2} + (- 2 - 4)^{2} = 16 + 36$
1 mark
(ii)(a) $A C = 52 = 213$ units
1 mark
(ii)(b) From the graph, A(3, 4) and B(3, 2), with $m : n = 1 : 3$ . By the section formula, the point is $(\frac{1 \times 3 + 3 \times 3}{1 + 3}, \frac{1 \times 2 + 3 \times 4}{1 + 3})$
1 mark
(ii)(b) $= (\frac{12}{4}, \frac{14}{4}) = (3, \frac{7}{2})$
1 mark
(iii) Reading from the graph, D is at $(- 2, - 5)$ .
1 mark

Answer: (i) (−1, 2); (ii)(a) $52 = 213$ units; (ii)(b) $(3, 7/2)$ ; (iii) D(−2, −5)

Where the marks are

(i) Mid-point of FG = (−1, 2) (1 mark)
(ii)(a) Correct substitution in the distance formula (1 mark)
(ii)(a) $A C = 213$ (1 mark)
(ii)(b) Correct substitution in the section formula (1 mark)
(ii)(b) Point $(3, 7/2)$ (1 mark)
(iii) D(−2, −5) (1 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 paper2024 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 2024: question 36

Hints

Step-by-step solution

Revise the chapter

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