MathsCBSE Class 10 · Standard · Board exam 2025, set 30/4/1 · 3 marks · Short answer

CBSE Class 10 Mathematics Standard 2025: question 31

Set in the CBSE Class 10 board exam of 2025, 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

Rectangle ABCD circumscribes the circle of radius 10 cm. Prove that ABCD is a square. Hence, find the perimeter of ABCD.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2025 board paper, set 30/4/1.

Hints

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

Hint 1

Tangents from a point outside a circle to it are equal in length.

Hint 2

Add the four tangent equalities, then use 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.

Figure: rectangle ABCD with a circle (centre O, radius 10 cm) touching AB at P, BC at Q, CD at R and DA at S.
½ mark
Tangents drawn from an external point are equal: $A P = A S$ , $B P = B Q$ , $C R = C Q$ , $D R = D S$ .
1 mark
Adding: $A P + B P + C R + D R = A S + B Q + C Q + D S \Rightarrow A B + C D = A D + B C$ ...(i)
½ mark
In rectangle ABCD, $A B = C D$ and $B C = A D$ , so (i) gives $2 A B = 2 A D \Rightarrow A B = A D$ . A rectangle with adjacent sides equal is a square, so ABCD is a square.
½ mark
Side of square = distance between opposite tangent points = diameter $= 2 \times 10 = 20$ cm. Perimeter $= 4 \times 20 = 80$ cm.
½ mark

Answer: ABCD is a square; perimeter = 80 cm

Where the marks are

Correct figure (½ mark)
Equal tangent lengths from each vertex (1 mark)
Adds to get $A B + C D = A D + B C$ (½ mark)
Uses rectangle property to get $A B = A D$ , hence square (½ mark)
Side = 20 cm, perimeter = 80 cm (½ mark)

Revise the chapter

Chapter: Circles (notes, formulas, where marks are lost and every past question)
Topic: Tangent and Secant of a Circle

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 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.

CBSE Class 10 Mathematics Standard 2025: question 31

Hints

Step-by-step solution

Revise the chapter

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