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Prove that the angle… (Class 10 Maths Basic 2024, Q27b)

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MathsCBSE Class 10 · Basic · Board exam 2024, set 430/2/1 · 3 marks · Short answer

CBSE Class 10 Mathematics Basic 2024: question 27(b)

Set in the CBSE Class 10 board exam of 2024, paper 430/2/1, section C, question 27(b), for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q27(b) · 3 marks · Short answerCircles

Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line segment joining the points of contact at the centre.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2024 board paper, set 430/2/1.

Hints

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

Hint 1

Join the centre to both points of contact.

Hint 2

What is the angle between a tangent and the radius at the point of contact?

Hint 3

Use the angle sum of quadrilateral OQPR.

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. Figure: a circle with centre O. PQ and PR are tangents from an external point P touching the circle at Q and R. Join OQ and OR. To prove: .

    ½ mark

  2. and [radius ⟂ tangent at the point of contact].

    1 mark

  3. In quadrilateral OQPR, the angle sum is : , so .

    1 mark

  4. Hence . The angle between the tangents is supplementary to the angle subtended by QR at the centre.

    ½ mark

Answer: (proved)

Where the marks are

  • Figure with given and to prove (½ mark)
  • Both angles between radius and tangent are (1 mark)
  • Uses the quadrilateral angle sum of (1 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 430/2/1 (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.