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PA and PB are… (Class 10 Maths Basic SQP 2024-25, Q30A)

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MathsCBSE Class 10 · Basic · Sample paper, session 2024-25 · 3 marks · Short answer

CBSE Class 10 Mathematics Basic sample paper 2024-25: question 30(A)

From the official CBSE sample question paper for 2024-25, 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, PA and PB are tangents to a circle centred at O. Prove that (i) OP bisects (ii) OP is the right bisector of AB.

Figure in the paper: A circle with centre O. External point P with tangents PA and PB touching the circle at A and B. Chord AB is drawn, and the line OP meets AB at Q. Radii OA and OB are drawn. See it in the official paper.

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2024-25.

Hints

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

Hint 1

Tangents drawn from an external point are equal, and radii are equal. Look for congruent triangles.

Hint 2

For (ii), compare triangles AQP and BQP. The two angles at Q form a linear pair.

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. (i) In and :

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

    So [SSS]. Hence [CPCT], i.e. OP bisects .

    1 mark

  2. (ii) In and :

    • [equal tangents]
    • [from (i)]
    • [common]

    So [SAS]. Hence and [CPCT].

    1 mark

  3. Since AQB is a straight line, , so .
    Thus OP is perpendicular to AB and bisects it at Q, so OP is the right (perpendicular) bisector of AB.

    1 mark

Answer: OP bisects ∠APB, and OP is the perpendicular bisector of AB (proved).

Where the marks are

  • , giving (1 mark)
  • , giving and equal angles at Q (1 mark)
  • Linear pair gives , so OP is the right bisector of AB (1 mark)

Revise the chapter

  • Chapter: Circles (notes, formulas, where marks are lost and every past question)
Practise this chapterBoard-style questions from Circles, one at a time, marked against the answer key and marking points.Back to the full paperSample paper 2024-25 (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.