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Altitudes CE and AD of… (Class 10 Maths Basic 2024, Q34a)

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MathsCBSE Class 10 · Basic · Board exam 2024, set 430/2/1 · 5 marks · Long answer

CBSE Class 10 Mathematics Basic 2024: question 34(a)

Set in the CBSE Class 10 board exam of 2024, paper 430/2/1, section D, question 34(a), for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q34(a) · 5 marks · Long answerTriangles

In the given figure, altitudes CE and AD of ABC intersect each other at the point P. Show that:

  1. (i)

    AEP CDP

    1 mark
  2. (ii)

    ABD CBE

    2 marks
  3. (iii)

    AEP ADB

    2 marks

Figure in the paper: Triangle ABC with A at bottom-left, B at bottom-right, C at top. Altitude CE drops from C to base AB at foot E (right angle marked at E). Altitude AD drops from A to side BC at foot D (right angle marked at D). The two altitudes intersect at point P inside the triangle. See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2024 board paper, set 430/2/1.

Hints

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

Hint 1

Each altitude makes a angle with the side it meets.

Hint 2

For each pair of triangles, find one right angle in each and one other shared or equal angle (common or vertically opposite).

Hint 3

Two pairs of equal angles prove similarity by the AA criterion. Keep the vertex order correct.

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. (i) In and : [vertically opposite angles]

    ½ mark

  2. (i) [CE and AD are altitudes]. So [AA similarity].

    ½ mark

  3. (ii) In and : [common angle B]

    ½ mark

  4. (ii) [AD ⟂ BC, CE ⟂ AB]

    1 mark

  5. (ii) So [AA similarity].

    ½ mark

  6. (iii) In and : [common angle A, since P lies on AD and E lies on AB]

    ½ mark

  7. (iii) [altitudes]

    1 mark

  8. (iii) So [AA similarity].

    ½ mark

Answer: (i) ; (ii) ; (iii) , all by AA similarity

Where the marks are

  • (i) Vertically opposite angles at P (½ mark)
  • (i) Right angles at E and D, so similar by AA (½ mark)
  • (ii) Common angle B (½ mark)
  • (ii) (1 mark)
  • (ii) Concludes similarity by AA (½ mark)
  • (iii) Common angle A (½ mark)
  • (iii) (1 mark)
  • (iii) Concludes similarity by AA (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 paper2024 board paper, set 430/2/1 (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.