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A quadrilateral ABCD is… (Class 10 Maths Basic 2023, Q28a)

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MathsCBSE Class 10 · Basic · Board exam 2023, set 430/2/1 · 3 marks · Short answer

CBSE Class 10 Mathematics Basic 2023: question 28(a)

Set in the CBSE Class 10 board exam of 2023, paper 430/2/1, section C, question 28(a), for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q28(a) · 3 marks · Short answerCircles

A quadrilateral ABCD is drawn to circumscribe a circle, as shown in the figure. Prove that .

Figure in the paper: Quadrilateral ABCD with a circle inscribed inside it, touching all four sides. Vertices A (bottom left), B (bottom right), C (top right), D (top left). See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2023 board paper, set 430/2/1.

Hints

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

Hint 1

Mark the points where the circle touches each side.

Hint 2

From each vertex two tangents are drawn to the circle. What do you know about their lengths?

Hint 3

Write AB and CD as sums of two tangent pieces and regroup them.

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. Let the circle touch AB, BC, CD, DA at P, Q, R, S respectively. Tangents drawn from an external point to a circle are equal, so , , , .

    1 mark

  2. ½ mark

  3. [using the equal tangents]

    ½ mark

  4. ½ mark

  5. . Hence .

    ½ mark

Answer: AB + CD = AD + BC (proved using equal tangents from external points).

Where the marks are

  • Names points of contact and writes the four pairs of equal tangents (1 mark)
  • Splits AB + CD into tangent segments (1/2 mark)
  • Replaces with equal tangents (1/2 mark)
  • Regroups the terms (1/2 mark)
  • Concludes AD + BC (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 paper2023 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.