MathsCBSE Class 10 · Standard · Board exam 2023, set 30/1/1 · 3 marks · Short answer

CBSE Class 10 Mathematics Standard 2023: question 28

Set in the CBSE Class 10 board exam of 2023, paper 30/1/1, section C, question 28, for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q28 · 3 marks · Short answerCoordinate Geometry

If $(- 5, 3)$ and $(5, 3)$ are two vertices of an equilateral triangle, then find co-ordinates of the third vertex, given that origin lies inside the triangle. (Take $3 = 1.7$ )

Source: official question paper from cbse.gov.in (PDF). See it in the full 2023 board paper, set 30/1/1.

Hints

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

Hint 1

Every side of an equilateral triangle has the same length. Find $A B$ first.

Hint 2

Equate $A C^{2}$ and $B C^{2}$ to find $x$ , then use $A C = 10$ to find $y$ . Decide the sign so that the origin is inside.

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.

Let $A (- 5, 3)$ , $B (5, 3)$ and let the third vertex be $C (x, y)$ .
Side $A B = (5 + 5)^{2} + 0^{2} = 10$ .
Since the triangle is equilateral, $A C = B C = A B = 10$ .
1 mark
From $A C^{2} = B C^{2}$ :
$(x + 5)^{2} + (y - 3)^{2} = (x - 5)^{2} + (y - 3)^{2}$
$\Rightarrow 20 x = 0 \Rightarrow x = 0$ .
1 mark
From $A C^{2} = 100$ with $x = 0$ :
$25 + (y - 3)^{2} = 100 \Rightarrow (y - 3)^{2} = 75 \Rightarrow y - 3 = \pm 53$ .
½ mark
$A B$ lies on the line $y = 3$ , above the origin. For the origin to lie inside the triangle, $C$ must lie below the x-axis, so we take the negative sign.
$y = 3 - 53 = 3 - 5 (1.7) = 3 - 8.5 = - 5.5$ .
The third vertex is $(0, - 5.5)$ .
½ mark

Answer: $(0, - 5.5)$ , i.e. $(0, 3 - 53)$

Where the marks are

Finds $A B = 10$ and sets $A C = B C = 10$ (1 mark)
Gets $x = 0$ (1 mark)
Gets $(y - 3)^{2} = 75$ (½ mark)
Chooses the negative root and gets $(0, - 5.5)$ (½ mark)

Revise the chapter

Chapter: Coordinate Geometry (notes, formulas, where marks are lost and every past question)

Practise this chapterBoard-style questions from Coordinate Geometry, one at a time, marked against the answer key and marking points.Back to the full paper2023 board paper, set 30/1/1 (Standard): every question, by section.

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CBSE Class 10 Mathematics Standard 2023: question 28

Hints

Step-by-step solution

Revise the chapter

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