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Prove that the… (Class 10 Maths Basic 2024, Q33b)

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MathsCBSE Class 10 · Basic · Board exam 2024, set 430/3/1 · 5 marks · Long answer

CBSE Class 10 Mathematics Basic 2024: question 33(b)

Set in the CBSE Class 10 board exam of 2024, paper 430/3/1, section D, question 33(b), for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q33(b) · 5 marks · Long answerCircles

Prove that the parallelogram circumscribing a circle is a rhombus. Also, find area of the rhombus, if radius of circle is cm and length of one side of the rhombus is cm.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2024 board paper, set 430/3/1.

Hints

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

Hint 1

Use: lengths of tangents drawn from an external point to a circle are equal.

Hint 2

Add the four equalities and group them into the sides of the parallelogram.

Hint 3

For area, the radius to each point of contact is the height of a triangle with a side of the rhombus as base.

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: Parallelogram ABCD circumscribes a circle with centre O, touching AB, BC, CD, DA at P, Q, R, S respectively; OA, OB and OP are joined.

    1 mark

  2. Tangents from an external point are equal: , , , .

    1 mark

  3. Adding:

    ½ mark

  4. ½ mark

  5. In a parallelogram and , so . Adjacent sides equal, hence ABCD is a rhombus.

    1 mark

  6. [tangent ⟂ radius], cm. The four triangles AOB, BOC, COD, DOA each have base 10 cm and height 3 cm. Area of rhombus cm²

    1 mark

Answer: ABCD is a rhombus; area = 60 cm²

Where the marks are

  • Correct labelled figure (1 mark)
  • Equal tangent lengths from each vertex (1 mark)
  • Adding the equalities (½ mark)
  • (½ mark)
  • Using opposite sides equal to get , so rhombus (1 mark)
  • Area cm² (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 paper2024 board paper, set 430/3/1 (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.