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QR/QS = QT/PR and ∠ 1 =… (Class 10 Maths Basic 2023, Q35b)

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MathsCBSE Class 10 · Basic · Board exam 2023, set 430/6/1 · 5 marks · Long answer

CBSE Class 10 Mathematics Basic 2023: question 35(b)

Set in the CBSE Class 10 board exam of 2023, paper 430/6/1, section D, question 35(b), for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q35(b) · 5 marks · Long answerTriangles

In the given figure, and . Prove that PQS TQR.

Figure in the paper: Triangle QTR with vertex T at the top and base QR. Point P lies on side QT and point S lies on base QR, with PS and PR joined. Angle 1 is marked at Q (between QP and QS) and angle 2 is marked at R (between RS/RP region near base). Points are labelled Q (bottom left), R (bottom right), S (on base), T (top), P (on QT). See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2023 board paper, set 430/6/1.

Hints

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

Hint 1

What does tell you about the sides of ?

Hint 2

Replace PR by PQ in the given ratio, then flip both sides.

Hint 3

Angle Q is common to both triangles. Which sides enclose it in each triangle?

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 , , i.e. .
    So [sides opposite equal angles are equal].

    1 mark

  2. Given . Taking reciprocals gives .
    Since , this becomes ...(i)

    2 marks

  3. In and :

    • [common angle Q]
    • [from (i)]
    1½ marks

  4. So [SAS similarity criterion]. Proved.

    ½ mark

Answer: by SAS similarity.

Where the marks are

  • Uses to get (1 mark)
  • Rearranges the given ratio and substitutes PR = PQ to get (2 marks)
  • Identifies the common angle Q and the proportional sides (1½ marks)
  • Concludes similarity by SAS (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 paper2023 board paper, set 430/6/1 (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.