MathsCBSE Class 10 · Basic · Sample paper, session 2023-24 · 4 marks · Case-based

CBSE Class 10 Mathematics Basic sample paper 2023-24: question 37

From the official CBSE sample question paper for 2023-24, 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

Alia and Shagun are friends living on the same street in Patel Nagar. Shagun's house is at the intersection of one street with another street on which there is a library. They both study in the same school and that is not far from Shagun's house. Suppose the school is situated at the point O, i.e., the origin, Alia's house is at A. Shagun's house is at B and library is at C. Based on the above information, answer the following questions.

(i)
How far is Alia's house from Shagun's house?
1 mark
(ii)
How far is the library from Shagun's house?
1 mark
(iii)
Show that for Shagun, school is farther compared to Alia's house and library.
2 marks
OR
(iii)
Show that Alia's house, Shagun's house and library form an isosceles right triangle.
2 marks

Figure in the paper: Coordinate plane with X and X' axes and Y and Y' axes, origin O. Points plotted: A at (2, 3), B at (2, 1), C at (4, 1). Axes are marked at intervals of 1 from -1 to 4 on X and from -1 to 4 on Y. See it in the official paper.

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2023-24.

Hints

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

Hint 1

Read the coordinates of O, A, B and C from the graph first.

Hint 2

Use $(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}$ for each pair of points.

Hint 3

For the triangle, check whether two sides are equal and whether the squares of the two shorter sides add up to the square of the longest.

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.

From the figure: school $O (0, 0)$ , Alia's house $A (2, 3)$ , Shagun's house $B (2, 1)$ , library $C (4, 1)$ . Distance formula: $d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}$ .
(i) $A B = (2 - 2)^{2} + (1 - 3)^{2} = 0 + (- 2)^{2}$
½ mark
$= 4 = 2$ units. Alia's house is 2 units from Shagun's house.
½ mark
(ii) $C B = (2 - 4)^{2} + (1 - 1)^{2} = (- 2)^{2} + 0$
½ mark
$= 4 = 2$ units. The library is 2 units from Shagun's house.
½ mark
(iii)(a) $O B = (2 - 0)^{2} + (1 - 0)^{2} = 4 + 1 = 5$ units
1 mark
From (i) and (ii), $A B = 2$ units and $C B = 2$ units, i.e. $A B = C B = 2 = 4$ .
½ mark
Since $5 > 4$ , $O B > A B$ and $O B > C B$ . Hence, for Shagun, the school is farther than Alia's house and the library.
½ mark
(iii)(b) $A C = (2 - 4)^{2} + (3 - 1)^{2} = 4 + 4 = 8 = 22$ units, so $A C^{2} = 8$ .
1 mark
$A B = B C = 2$ units, so the triangle is isosceles.
½ mark
$A B^{2} + B C^{2} = 4 + 4 = 8 = A C^{2}$ , so by the converse of Pythagoras theorem $∠ B = 9 0^{\circ}$ . Hence A, B and C form an isosceles right triangle.
½ mark

Answer: (i) 2 units; (ii) 2 units; (iii)(a) OB = √5 units > AB = CB = 2 units, so the school is farther; (iii)(b) AB = BC = 2, AC = 2√2 and AB² + BC² = AC², so it is an isosceles right triangle

Where the marks are

Substitutes into the distance formula for AB (½ mark)
Gets AB = 2 units (½ mark)
Substitutes into the distance formula for CB (½ mark)
Gets CB = 2 units (½ mark)
(iii)(a) Finds OB = √5 units (1 mark)
Uses AB = CB = 2 units (½ mark)
Concludes OB is greater, so the school is farther (½ mark)
(iii)(b) Finds AC = 2√2, i.e. AC² = 8 (1 mark)
Notes AB = BC = 2 (½ mark)
Shows AB² + BC² = AC², so it is an isosceles right triangle (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 paperSample paper 2023-24 (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 sample paper 2023-24: question 37

Hints

Step-by-step solution

Revise the chapter

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