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Prove that a rectangle… (Class 10 Maths Basic 2025, Q29)

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MathsCBSE Class 10 · Basic · Board exam 2025, set 430/1/1 · 3 marks · Short answer

CBSE Class 10 Mathematics Basic 2025: question 29

Set in the CBSE Class 10 board exam of 2025, paper 430/1/1, section C, question 29, for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q29 · 3 marks · Short answerCircles

Prove that a rectangle circumscribing a circle is a square.

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

Hints

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

Hint 1

Mark the points where the circle touches each side, and use the fact that tangents from a point are equal.

Hint 2

Add the four tangent equalities, then use the fact that opposite sides of a rectangle are equal.

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: rectangle ABCD circumscribing a circle with centre O. The sides AB, BC, CD, DA touch the circle at P, Q, R, S respectively.

    ½ mark

  2. Tangents drawn from an external point to a circle are equal, so , , and . Adding: , i.e. .

    1 mark

  3. In a rectangle, opposite sides are equal: and . So .

    1 mark

  4. Adjacent sides of rectangle ABCD are equal, so ABCD is a square.

    ½ mark

Answer: AB + CD = AD + BC with AB = CD and AD = BC gives AB = AD, so ABCD is a square

Where the marks are

  • Draws a correct figure (½ mark)
  • Uses equal tangents to get (1 mark)
  • Uses opposite sides of the rectangle to show (1 mark)
  • Concludes ABCD is a square (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 430/1/1 (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.