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PQ ∥ RS. Prove that OP ×… (Class 10 Maths Basic 2024, Q23a)

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MathsCBSE Class 10 · Basic · Board exam 2024, set 430/4/1 · 2 marks · Very short answer

CBSE Class 10 Mathematics Basic 2024: question 23(a)

Set in the CBSE Class 10 board exam of 2024, paper 430/4/1, section B, question 23(a), for 2 marks. Try it first; the hints and our step-by-step solution are below.

Q23(a) · 2 marks · Very short answerTriangles

In the given figure, . Prove that .

Figure in the paper: Two lines PR and QS intersect at O. P and Q are joined by segment PQ, and R and S by segment RS, with PQ parallel to RS (arrow marks). P is on one arm and R on the other arm of the crossing at O; Q and S lie on the other pair of arms. See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2024 board paper, set 430/4/1.

Hints

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

Hint 1

Look for two triangles with vertex O that might be similar.

Hint 2

Use vertically opposite angles at O and alternate angles from PQ ∥ RS, then write the proportional sides.

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 : [vertically opposite angles].

    ½ mark

  2. [alternate interior angles, since and PS is a transversal].

    ½ mark

  3. Therefore [AA similarity criterion].

    ½ mark

  4. Corresponding sides are proportional, so . Cross-multiplying gives . Hence proved.

    ½ mark

Answer: △POQ ~ △SOR (AA), so OP/OS = OQ/OR, i.e. OP × OR = OQ × OS.

Where the marks are

  • Vertically opposite angles are equal (½ mark)
  • Alternate interior angles are equal because PQ ∥ RS (½ mark)
  • Concludes △POQ ~ △SOR by AA (½ mark)
  • Writes the ratio of corresponding sides and cross-multiplies (½ mark)

Revise the chapter

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 430/4/1 (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.