MathsCBSE Class 10 · Basic · Board exam 2023, set 430/5/1 · 2 marks · Very short answer

CBSE Class 10 Mathematics Basic 2023: question 23

Set in the CBSE Class 10 board exam of 2023, paper 430/5/1, section B, question 23, for 2 marks. Try it first; the hints and our step-by-step solution are below.

Q23 · 2 marks · Very short answerCircles

In the given figure, tangents AB and AC are drawn to a circle centred at O. If $∠ O A B = 6 0^{\circ}$ and OB = 5 cm, find lengths OA and AC.

Figure in the paper: A circle with centre O. Two tangents AB and AC are drawn from external point A, touching the circle at B and C. OB = 5 cm is marked as a radius. The angle OAB = $6 0^{\circ}$ is marked at A. O, A and the line joining them are shown. See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2023 board paper, set 430/5/1.

Hints

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

Hint 1

What angle does a tangent make with the radius at the point of contact?

Hint 2

In right triangle OBA, OB is opposite the 60° angle — use sin for OA and tan for AB.

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.

$O B ⊥ A B$ [tangent ⟂ radius at the point of contact], so $△ O B A$ is right-angled at B. In it, $sin 6 0^{\circ} = \frac{O B}{O A} \Rightarrow \frac{3}{2} = \frac{5}{O A}$
½ mark
$O A = \frac{10}{3} = \frac{10 3}{3}$ cm
½ mark
$tan 6 0^{\circ} = \frac{O B}{A B} \Rightarrow 3 = \frac{5}{A B}$
½ mark
$A B = \frac{5}{3} = \frac{5 3}{3}$ cm, and $A C = A B = \frac{5 3}{3}$ cm [tangents from an external point are equal]
½ mark

Answer: $O A = \frac{10 3}{3}$ cm, $A C = \frac{5 3}{3}$ cm

Where the marks are

Uses $sin 6 0^{\circ} = O B / O A$ in right $△ O B A$ (½ mark)
$O A = \frac{10 3}{3}$ cm (½ mark)
Uses $tan 6 0^{\circ} = O B / A B$ (½ mark)
$A C = A B = \frac{5 3}{3}$ cm (½ mark)

Revise the chapter

Chapter: Circles (notes, formulas, where marks are lost and every past question)
Topic: Solving Problems Using Tangent Properties

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 430/5/1 (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.

CBSE Class 10 Mathematics Basic 2023: question 23

Hints

Step-by-step solution

Revise the chapter

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