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Prove that the tangent… (Class 10 Maths Basic 2025, Q23)

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MathsCBSE Class 10 · Basic · Board exam 2025, set 430/2/1 · 2 marks · Very short answer

CBSE Class 10 Mathematics Basic 2025: question 23

Set in the CBSE Class 10 board exam of 2025, paper 430/2/1, section B, question 23, for 2 marks. Try it first; the hints and our step-by-step solution are below.

Q23 · 2 marks · Very short answerCircles

Prove that the tangent drawn at any point of the circle is perpendicular to the radius through the point of contact.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2025 board paper, set 430/2/1.

Hints

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

Hint 1

Take a point on the tangent other than the point of contact and join it to the centre.

Hint 2

Where does this point lie relative to the circle? Compare its distance from O with the radius.

Hint 3

Which segment from a point to a line is the shortest?

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: circle with centre O, line touching it at A, radius OA drawn, and another point B on joined to O. Given: is the tangent at A. To prove: .

    ½ mark

  2. Proof: Take any point B on other than A and join OB.

    ½ mark

  3. Since a tangent meets the circle at only one point, B lies outside the circle, so radius, i.e. .

    ½ mark

  4. Thus OA is the shortest of all segments from O to points of . The shortest distance from a point to a line is the perpendicular, so . Hence proved.

    ½ mark

Answer: OA is the shortest distance from O to the tangent, hence OA ⟂ tangent.

Where the marks are

  • Correct figure with given/to prove (½ mark)
  • Takes another point B on the tangent (½ mark)
  • Shows B outside the circle so OB > OA (½ mark)
  • Concludes OA ⟂ l since perpendicular is the shortest distance (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 paper2025 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.