In the given figure, a circle is inscribed in a quadrilateral ABCD in which ∠ B = 90°. If AD = 17 cm, AB = 20 cm and DS = 3 cm, then find the radius of the circle.

MathsCBSE Class 10 · Standard · Board exam 2023, set 30/1/1 · 3 marks · Short answer

CBSE Class 10 Mathematics Standard 2023: question 29(b)

Set in the CBSE Class 10 board exam of 2023, paper 30/1/1, section C, question 29(b), for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q29(b) · 3 marks · Short answerCircles

In the given figure, a circle is inscribed in a quadrilateral ABCD in which $∠ B = 9 0^{\circ}$ . If $A D = 17$ cm, $A B = 20$ cm and $D S = 3$ cm, then find the radius of the circle.

Figure in the paper: Quadrilateral ABCD with a circle inscribed, centre O. Touch points: R on AD, S on DC, P on CB, Q on AB. Angle B = 90° is marked at B. Radii OP and OQ are marked r each. Given lengths: AD = 17 cm, AB = 20 cm, DS = 3 cm. See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2023 board paper, set 30/1/1.

Hints

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

Hint 1

Tangent lengths from the same external point are equal. Start at D, then move to A.

Hint 2

Find $QB$ . Then look at the figure $O QB P$ : it has three right angles and two equal adjacent sides.

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.

$D R = D S = 3$ cm [tangents from an external point are equal].
½ mark
$A R = A D - D R = 17 - 3 = 14$ cm.
1 mark
$A Q = A R = 14$ cm [tangents from A].
½ mark
$QB = A B - A Q = 20 - 14 = 6$ cm.
½ mark
In quadrilateral $O QB P$ :
- $∠ O QB = ∠ O P B = 9 0^{\circ}$ [tangent $⊥$ radius]
- $∠ B = 9 0^{\circ}$
- $O Q = O P = r$
So $O QB P$ is a square, and $r = O P = QB = 6$ cm.
The radius is 6 cm.
½ mark

Answer: Radius = 6 cm

Where the marks are

$D R = D S = 3$ cm (½ mark)
$A R = 14$ cm (1 mark)
$A Q = A R = 14$ cm (½ mark)
$QB = 6$ cm (½ mark)
$O QB P$ is a square, so $r = 6$ cm (½ 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 paper2023 board paper, set 30/1/1 (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.

CBSE Class 10 Mathematics Standard 2023: question 29(b)

Hints

Step-by-step solution

Revise the chapter

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