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CD is the perpendicular… (Class 10 Maths 2023, Q28a)

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MathsCBSE Class 10 · Standard · Board exam 2023, set 30/5/1 · 3 marks · Short answer

CBSE Class 10 Mathematics Standard 2023: question 28(a)

Set in the CBSE Class 10 board exam of 2023, paper 30/5/1, section C, question 28(a), for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q28(a) · 3 marks · Short answerTriangles

In the given figure, CD is the perpendicular bisector of AB. EF is perpendicular to CD. AE intersects CD at G. Prove that .

Figure in the paper: Points A and B joined by segment AB; D is the foot on AB (right angle marked at D) and CD is perpendicular to AB, with C on the left. Point E lies on segment CB; EF is perpendicular to CD with F on CD. Segment AE meets CD at G. See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2023 board paper, set 30/5/1.

Hints

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

Hint 1

Look for two pairs of similar right-angled triangles: one around G and one with common vertex C.

Hint 2

Both similarities give a ratio involving EF; D being the midpoint of AB links AD and BD.

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. In and : and (vertically opposite angles). So (AA similarity).
    ……(i)

    1 mark

  2. In and : and (common angle). So (AA similarity).

    1 mark

  3. CD is the perpendicular bisector of AB, so . Hence ……(ii)

    ½ mark

  4. From (i) and (ii): . Hence proved.

    ½ mark

Answer: Using ΔEFG ~ ΔADG and ΔEFC ~ ΔBDC with AD = BD, CF/CD = FG/DG (proved).

Where the marks are

  • Proves ΔEFG ~ ΔADG and writes EF/AD = FG/DG (1 mark)
  • Proves ΔEFC ~ ΔBDC and writes EF/BD = CF/CD (1 mark)
  • Uses BD = AD (D is the midpoint of AB) to get EF/AD = CF/CD (½ mark)
  • Combines to get CF/CD = FG/DG (½ mark)

Revise the chapter

  • Chapter: Triangles (notes, formulas, where marks are lost and every past question)
Practise this chapterBoard-style questions from Triangles, one at a time, marked against the answer key and marking points.Back to the full paper2023 board paper, set 30/5/1 (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.