MathsCBSE Class 10 · Standard · Board exam 2023, set 30/6/1 · 5 marks · Long answer

CBSE Class 10 Mathematics Standard 2023: question 32(B)

Set in the CBSE Class 10 board exam of 2023, paper 30/6/1, section D, question 32(B), for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q32(B) · 5 marks · Long answerCircles

A circle touches the side BC of a $△ A B C$ at a point P and touches AB and AC when produced at Q and R respectively. Show that $A Q = \frac{1}{2}$ (Perimeter of $△ A B C$ ).

Figure in the paper: Triangle ABC with apex A at top; side BC is horizontal at the bottom with P on BC. A circle lies below BC touching BC at P, and touches the extended sides AB (at Q, beyond B) and AC (at R, beyond C). See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2023 board paper, set 30/6/1.

Hints

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

Hint 1

List the pairs of equal tangents drawn from A, B and C.

Hint 2

Write AQ + AR as AB + BQ + AC + CR, then replace BQ and CR with segments of BC.

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.

Tangents from an external point to a circle are equal in length: AQ = AR (from A), BQ = BP (from B), CP = CR (from C).
1 mark
So $2 A Q = A Q + A R$ .
1 mark
Since Q lies on AB produced and R on AC produced, $A Q = A B + B Q$ and $A R = A C + C R$ . So $2 A Q = A B + B Q + A C + C R$ .
½ mark
Replacing BQ by BP and CR by CP: $2 A Q = A B + A C + (B P + C P)$ .
½ mark
$B P + P C = B C$ , so $2 A Q = A B + A C + B C$ .
1 mark
Therefore $A Q = \frac{1}{2} (A B + B C + C A) = \frac{1}{2}$ (Perimeter of $△ A B C$ ).
1 mark

Answer: $A Q = \frac{1}{2} (A B + B C + C A)$ , i.e. half the perimeter of $△ A B C$ .

Where the marks are

Uses equal tangents AQ = AR, BQ = BP, CP = CR (1 mark)
Writes 2AQ = AQ + AR (1 mark)
Splits into AB + BQ + AC + CR (½ mark)
Replaces with BP + CP (½ mark)
Uses BP + CP = BC (1 mark)
Concludes AQ = ½ perimeter (1 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 paper2023 board paper, set 30/6/1 (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 32(B)

Hints

Step-by-step solution

Revise the chapter

More questions from Circles