Skip to content

Observe the figures given… (Class 10 Maths Basic 2023, Q38)

questionbuddy.in/q/cbse-10-maths-2023-430-4-1-q38-observe-figures-carefully

MathsCBSE Class 10 · Basic · Board exam 2023, set 430/4/1 · 4 marks · Case-based

CBSE Class 10 Mathematics Basic 2023: question 38

Set in the CBSE Class 10 board exam of 2023, paper 430/4/1, section E, question 38, for 4 marks. Try it first; the hints and our step-by-step solution are below.

Q38 · 4 marks · Case-basedCompetency-basedTriangles

Observe the figures given below carefully and answer the questions:

  1. (i)

    Name the figure(s) wherein two figures are similar.

    1 mark
  2. (ii)

    Name the figure(s) wherein the figures are congruent.

    1 mark
  3. (iii)(a)

    Prove that congruent triangles are also similar but not the converse.

    2 marks

  4. OR

    (iii)(b)

    What more is least needed for two similar triangles to be congruent?

    2 marks

Figure in the paper: Figure A: A(i) is a quadrilateral ABCD with AD = 2 cm, DC = 2.4 cm, CB = 2.5 cm, AB = 1.2 cm; angles D = 85°, C = 70°, B = 100°, A = 105°. A(ii) is a quadrilateral PQRS with SR = 4.8 cm, RQ = 5 cm, QP = 2.4 cm, PS = 4 cm; angles S = 85°, R = 70°, Q = 100°, P = 105°. Figure B: B(iii) is a square ABCD with sides 3.5 cm on the top and bottom and 3 cm on the left and right, all angles right angles. B(iv) is a square PQRS with sides 3 cm on all four sides, all angles right angles. Figure C: C(v) is triangle ABC with AB = 2 cm, BC = 3 cm, AC = 4.5 cm. C(vi) is triangle PQR with PQ = 2 cm, QR = 3 cm, PR = 4.5 cm, with matching tick marks on corresponding sides. See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2023 board paper, set 430/4/1.

Hints

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

Hint 1

Two figures are similar when their corresponding angles are equal and their corresponding sides are in the same ratio. Check the ratios in each pair.

Hint 2

Congruent figures have exactly the same size and shape. Which pair has equal sides throughout?

Hint 3

For similar triangles, think about what the ratio of corresponding sides must be for them to be congruent.

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) Figure A: the corresponding angles are equal (85°, 70°, 100°, 105°). The corresponding sides are in the same ratio: . So A(i) and A(ii) are similar.

    ½ mark

  2. (i) Figure C: the triangles have equal corresponding sides (2 cm, 3 cm, 4.5 cm), so they are congruent and hence similar. Figure B is not similar, because .

    Answer: Figure A and Figure C.

    ½ mark

  3. (ii) Only Figure C has congruent figures: by SSS, since cm, cm and cm.

    1 mark

  4. (iii)(a) If , then by CPCT , , , and .

    1 mark

  5. (iii)(a) Hence the corresponding angles are equal and the corresponding sides are proportional, so the triangles are similar.

    ½ mark

  6. (iii)(a) The converse is not true. Take an equilateral triangle of side 2 cm and one of side 4 cm. All their angles are 60°, so they are similar, but their sides are not equal (). So similar triangles need not be congruent.

    ½ mark

  7. (iii)(b) Similar triangles already have equal angles and proportional sides. If at least one pair of corresponding sides is equal, the ratio of proportionality becomes 1, so all corresponding sides are equal and the triangles are congruent.

    So what is least needed is one pair of equal corresponding sides.

    2 marks

Answer: (i) Figure A and Figure C; (ii) Figure C; (iii)(a) congruent ⇒ equal angles and sides in ratio 1 ⇒ similar; the converse fails (e.g. equilateral triangles of sides 2 cm and 4 cm); (iii)(b) one pair of corresponding sides must be equal

Where the marks are

  • (i) Names Figure A (½ mark) and Figure C (½ mark)
  • (ii) Figure C (1 mark)
  • (iii)(a) Congruent ⇒ corresponding angles equal (1 mark)
  • (iii)(a) Hence similar (½ mark)
  • (iii)(a) Shows the converse fails, with a reason or counter-example (½ mark)
  • (iii)(b) At least one pair of corresponding sides must be equal (2 marks)

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 430/4/1 (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.