MathsCBSE Class 10 · Standard · Sample paper, session 2024-25 · 5 marks · Long answer

CBSE Class 10 Mathematics Standard sample paper 2024-25: question 33

From the official CBSE sample question paper for 2024-25, section D, question 33, for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q33 · 5 marks · Long answerCircles

Prove that the lengths of tangents drawn from an external point to a circle are equal.

Using above result, find the length $B C$ of $△ A B C$ . Given that, a circle is inscribed in $△ A B C$ touching the sides $A B$ , $B C$ and $C A$ at $R$ , $P$ and $Q$ respectively and $A B = 10$ cm, $A Q = 7$ cm, $C Q = 5$ cm.

Figure in the paper: Triangle ABC with A at the top, B bottom-left, C bottom-right. A circle is inscribed in the triangle, touching AB at R (left side), CA at Q (right side) and BC at P (bottom). See it in the official paper.

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2024-25.

Hints

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

Hint 1

For the proof, join the centre to the external point and to both points of contact. What angle does a radius make with a tangent?

Hint 2

Two right triangles that share a hypotenuse and have equal radii are congruent by RHS. For the triangle, pair up equal tangents from each vertex: $A R = A Q$ , $B R = B P$ , $C P = C Q$ .

Hint 3

Find $B R$ as $A B - A R$ , then add $B P$ and $P C$ .

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.

Given: A circle with centre $O$ . $P Q$ and $P R$ are two tangents drawn from an external point $P$ , touching the circle at $Q$ and $R$ .
To prove: $P Q = P R$ .
Construction: Join $O P$ , $O Q$ and $O R$ .
Figure: A circle with centre $O$ and a point $P$ outside it. Tangent segments $P Q$ and $P R$ are drawn, and the radii $O Q$ , $O R$ and the segment $O P$ are joined.
1 mark
Proof: In $△ O QP$ and $△ O R P$ :
- $∠ O QP = ∠ O R P = 9 0^{\circ}$ [tangent $⊥$ radius at the point of contact]
- $O Q = O R$ [radii of the same circle]
- $O P = O P$ [common hypotenuse]
1 mark
So $△ O QP ≅ △ O R P$ [RHS congruence rule].
Hence $P Q = P R$ [CPCT].
Therefore the lengths of the tangents drawn from an external point to a circle are equal.
1 mark
Application: Tangents from $A$ are $A R$ and $A Q$ . So $A R = A Q = 7$ cm.
½ mark
Tangents from $B$ are $B R$ and $B P$ . So $B P = B R = A B - A R = 10 - 7 = 3$ cm.
½ mark
Tangents from $C$ are $C P$ and $C Q$ . So $C P = C Q = 5$ cm.
½ mark
$B C = B P + P C = 3 + 5 = 8$ cm.
½ mark

Answer: Theorem proved (tangents from an external point are equal); BC = 8 cm

Where the marks are

Writes the given, to prove and construction, with a correct figure (1 mark)
Proves $△ O QP ≅ △ O R P$ using the right angles, equal radii and common side (RHS), and concludes $P Q = P R$ (2 marks)
States $A R = A Q = 7$ cm (½ mark)
Finds $B R = B P = 3$ cm (½ mark)
States $C P = C Q = 5$ cm (½ mark)
Gets $B C = 8$ cm (½ mark)

Revise the chapter

Chapter: Circles (notes, formulas, where marks are lost and every past question)
Topic: Lengths of Tangents from an External Point

Practise this chapterBoard-style questions from Circles, one at a time, marked against the answer key and marking points.Back to the full paperSample paper 2024-25 (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 sample paper 2024-25: question 33

Hints

Step-by-step solution

Revise the chapter

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