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PA, QB and RC are each… (Class 10 Maths 2024, Q34B)

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MathsCBSE Class 10 · Standard · Board exam 2024, set 30/1/1 · 5 marks · Long answer

CBSE Class 10 Mathematics Standard 2024: question 34(B)

Set in the CBSE Class 10 board exam of 2024, paper 30/1/1, section D, question 34(B), for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q34(B) · 5 marks · Long answerTriangles

In the given figure PA, QB and RC are each perpendicular to AC. If AP = x, BQ = y and CR = z, then prove that .

Figure in the paper: Points A and C lie on a horizontal base AC. B lies between A and C on the base. P is above A with PA perpendicular to AC (length x). Q lies on the segment PC with QB perpendicular to AC (length y). R is above C with RC perpendicular to AC (length z). Line AR passes through Q, and line PC also passes through Q... (Q is the intersection of PC and AR). See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2024 board paper, set 30/1/1.

Hints

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

Hint 1

Look for pairs of right triangles that share an angle at A or at C.

Hint 2

Use △PAC ~ △QBC and △RCA ~ △QBA to write y/x and y/z in terms of parts of AC.

Hint 3

Add the two ratios. Remember that AB + BC = 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. In and :

    • [given perpendiculars]
    • [common angle]

    So [AA similarity].

    1 mark

  2. Corresponding sides are proportional:

    ... (i)

    1 mark

  3. In and :

    • [common angle]

    So [AA similarity].

    1 mark

  4. Corresponding sides are proportional:

    ... (ii)

    1 mark

  5. Add (i) and (ii):

    [since B lies on AC]

    ½ mark

  6. Divide both sides by :

    . Hence proved.

    ½ mark

Answer: Proved: 1/x + 1/z = 1/y

Where the marks are

  • Proves (1 mark)
  • Obtains (1 mark)
  • Proves (1 mark)
  • Obtains (1 mark)
  • Adds the two ratios and uses (½ mark)
  • Concludes (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 30/1/1 (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.