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AB is a diameter of the circle… (Class 10 Maths 2024, Q30A)

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MathsCBSE Class 10 · Standard · Board exam 2024, set 30/1/1 · 3 marks · Short answer

CBSE Class 10 Mathematics Standard 2024: question 30(A)

Set in the CBSE Class 10 board exam of 2024, paper 30/1/1, section C, question 30(A), for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q30(A) · 3 marks · Short answerCircles

In the given figure, AB is a diameter of the circle with centre O. AQ, BP and PQ are tangents to the circle. Prove that .

Figure in the paper: A circle with centre O and diameter AB vertical. Tangent at A and tangent at B are horizontal lines. Point Q lies on the upper tangent to the right of A, and point P lies on the lower tangent to the right of B. Tangent PQ joins P and Q, and OQ and OP are drawn with the angle at O marked. See it in the official paper.

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

Hints

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

Hint 1

Mark the point R where PQ touches the circle and join it to O.

Hint 2

Tangents drawn from one external point are equal. Use this to find congruent triangles on each side of OR.

Hint 3

A, O, B lie on a straight line, so the angles at O add up to .

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.

  1. Let the tangent PQ touch the circle at R. Join OR.

    ½ mark

  2. In and :

    • [radii]
    • [tangents from an external point are equal]
    • [common]

    [SSS], so [CPCT] ...(1)

    1 mark

  3. In the same way, (OB = OR, PB = PR, OP common), so ...(2)

    ½ mark

  4. AOB is a diameter, so it is a straight line: .

    Using (1) and (2): .

    ½ mark

  5. , that is, . Hence proved.

    ½ mark

Answer: (proved)

Where the marks are

  • Takes R, the point of contact of PQ, and joins OR (½ mark)
  • Proves and gets (1 mark)
  • Proves and gets (½ mark)
  • Uses (½ mark)
  • Concludes (½ mark)

Revise the chapter

Practise this chapterBoard-style questions from Circles, one at a time, marked against the answer key and marking points.Back to the full paper2024 board paper, set 30/1/1 (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.