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Prove that the parallelogram… (Class 10 Maths 2024, Q31)

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MathsCBSE Class 10 · Standard · Board exam 2024, set 30/4/1 · 3 marks · Short answer

CBSE Class 10 Mathematics Standard 2024: question 31

Set in the CBSE Class 10 board exam of 2024, paper 30/4/1, section C, question 31, for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q31 · 3 marks · Short answerCircles

Prove that the parallelogram circumscribing a circle is a rhombus.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2024 board paper, set 30/4/1.

Hints

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

Hint 1

Mark the four points where the sides touch the circle.

Hint 2

Each vertex is an external point from which two tangents are drawn. What do you know about their lengths?

Hint 3

Add the four tangent equalities, then use the fact that opposite sides of a parallelogram 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: Draw parallelogram ABCD with all four sides touching a circle. The points of contact are P on AB, Q on BC, R on CD and S on DA.

    Given: ABCD is a parallelogram circumscribing the circle.

    To prove: ABCD is a rhombus.

    ½ mark

  2. Tangents drawn from an external point to a circle are equal in length. So:

    1 mark

  3. Adding these four equations:

    ½ mark

  4. In a parallelogram opposite sides are equal: and .

    So , which gives .

    ½ mark

  5. Therefore . A parallelogram with all sides equal is a rhombus, so ABCD is a rhombus. Hence proved.

    ½ mark

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

Where the marks are

  • Correct labelled figure (1/2 mark)
  • Writes the four equal-tangent relations (1 mark)
  • Adds them to get (1/2 mark)
  • Uses the opposite sides of the parallelogram to get (1/2 mark)
  • Concludes all sides are equal, so ABCD is a rhombus (1/2 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 30/4/1 (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.