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ABC is an isosceles… (Class 10 Maths Basic 2023, Q31)

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MathsCBSE Class 10 · Basic · Board exam 2023, set 430/5/1 · 3 marks · Short answer

CBSE Class 10 Mathematics Basic 2023: question 31

Set in the CBSE Class 10 board exam of 2023, paper 430/5/1, 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

ABC is an isosceles triangle with AB = AC, circumscribed about a circle. Prove that BC is bisected at E.

Figure in the paper: Isosceles triangle ABC with apex A at top and base BC horizontal. A circle is inscribed touching the sides at D (on AB), E (on BC) and F (on AC). E is the midpoint of BC where the circle touches the base. 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

Tangents drawn from the same external point to a circle are equal — list the three pairs.

Hint 2

Split AB and AC into tangent segments and use AB = AC.

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.

  1. The circle touches AB, BC, CA at D, E, F. Since tangents from an external point are equal: , , .

    1 mark

  2. Given

    ½ mark

  3. Using :

    ½ mark

  4. ½ mark

  5. [as and ]. Hence E is the mid-point of BC, i.e. BC is bisected at E.

    ½ mark

Answer: BE = EC, so BC is bisected at E.

Where the marks are

  • States equal tangent lengths AD = AF, BD = BE, CE = CF (1 mark)
  • Writes AB = AC as AD + DB = AF + FC (½ mark)
  • Substitutes AD = AF (½ mark)
  • Gets DB = FC (½ mark)
  • Concludes BE = EC (½ mark)

Revise the chapter

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.