Triangle is a very popular shape used in interior designing. The picture given above shows a cabinet designed by a famous interior designer. Here the largest triangle is represented by △ ABC and smallest one with shelf is represented by △ DEF. PQ is parallel to EF. (i) Show that △ DPQ ~ △ DEF. (ii) If DP=50 cm and PE=70 cm then find PQ/EF. (iii)(A) If 2AB=5DE and △ ABC ~ △ DEF then show that perimeter of △ ABCperimeter of △ DEF is constant. (iii)(B) If AM and DN are medians of triangles ABC and DEF respectively then prove that △ ABM ~ △ DEN.

MathsCBSE Class 10 · Standard · Sample paper, session 2024-25 · 4 marks · Case-based

CBSE Class 10 Mathematics Standard sample paper 2024-25: question 37

From the official CBSE sample question paper for 2024-25, section E, question 37, for 4 marks. Try it first; the hints and our step-by-step solution are below.

Q37 · 4 marks · Case-basedCompetency-basedTriangles

Triangle is a very popular shape used in interior designing. The picture given above shows a cabinet designed by a famous interior designer. Here the largest triangle is represented by $△ A B C$ and smallest one with shelf is represented by $△ D E F$ . $P Q$ is parallel to $E F$ .

(i)
Show that $△ D P Q \sim △ D E F$ .
1 mark
(ii)
If $D P = 50$ cm and $P E = 70$ cm then find $\frac{P Q}{E F}$ .
1 mark
(iii)(A)
If $2 A B = 5 D E$ and $△ A B C \sim △ D E F$ then show that $\frac{perimeter of △ A B C}{perimeter of △ D E F}$ is constant.
2 marks
OR
(iii)(B)
If $A M$ and $D N$ are medians of triangles $A B C$ and $D E F$ respectively then prove that $△ A B M \sim △ D E N$ .
2 marks

Figure in the paper: Top: a photograph of a triangular wooden shelf cabinet with shelves and decorative objects. Below: two triangles. Left triangle ABC is large with A at top, B bottom-left, C bottom-right. Right triangle DEF is small with D at top, E bottom-left, F bottom-right; P lies on DE and Q lies on DF, with segment PQ parallel to EF. See it in the official paper.

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2024-25.

Hints

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

Hint 1

$P Q ∥ E F$ gives corresponding angles, and the two triangles share $∠ D$ .

Hint 2

In similar triangles all corresponding sides are in the same ratio. Note that $D E = D P + P E$ .

Hint 3

For the medians, half of proportional sides are still in the same ratio. Combine this with the included angle.

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.

(i) In $△ D P Q$ and $△ D E F$ :
- $∠ D P Q = ∠ D E F$ [corresponding angles, $P Q ∥ E F$ ]
- $∠ P D Q = ∠ E D F$ [common angle]
So $△ D P Q \sim △ D E F$ [AA similarity].
1 mark
(ii) $D E = D P + P E = 50 + 70 = 120$ cm.
Since $△ D P Q \sim △ D E F$ , we have $\frac{D P}{D E} = \frac{P Q}{E F}$ .
½ mark
(ii) $\frac{P Q}{E F} = \frac{50}{120} = \frac{5}{12}$
½ mark
(iii)(a) $2 A B = 5 D E$ gives $\frac{A B}{D E} = \frac{5}{2}$ .
As $△ A B C \sim △ D E F$ , corresponding sides are proportional:
$\frac{A B}{D E} = \frac{B C}{E F} = \frac{A C}{D F} = \frac{5}{2}$
So $A B = \frac{5}{2} D E$ , $B C = \frac{5}{2} E F$ and $A C = \frac{5}{2} D F$ .
1 mark
(iii)(a) $\frac{perimeter of △ A B C}{perimeter of △ D E F} = \frac{\frac{5}{2} ( D E + E F + F D )}{D E + E F + F D} = \frac{5}{2}$
This is a constant.
1 mark
(iii)(b) Figure: $△ A B C$ with median $A M$ , where $M$ is the midpoint of $B C$ . $△ D E F$ with median $D N$ , where $N$ is the midpoint of $E F$ .
½ mark
(iii)(b) Since $△ A B C \sim △ D E F$ :
$\frac{A B}{D E} = \frac{B C}{E F} = \frac{B C /2}{E F /2} = \frac{B M}{E N}$
Also $∠ B = ∠ E$ [corresponding angles of similar triangles].
1 mark
(iii)(b) So $△ A B M \sim △ D E N$ [SAS similarity].
½ mark

Answer: (i) △DPQ ~ △DEF by AA; (ii) PQ/EF = 5/12; (iii)(a) ratio of perimeters = 5/2 (constant); (iii)(b) △ABM ~ △DEN by SAS

Where the marks are

(i) Gives two pairs of equal angles and concludes AA similarity (1 mark)
(ii) Finds $D E = 120$ cm and writes $D P / D E = P Q / E F$ (½ mark)
(ii) Gets $P Q / E F = 5/12$ (½ mark)
(iii)(a) Writes the proportional sides with ratio $5/2$ (1 mark)
(iii)(a) Shows the perimeter ratio $= 5/2$ , a constant (1 mark)
(iii)(b) Draws a correct figure with the medians (½ mark)
(iii)(b) Shows $A B / D E = B M / E N$ and $∠ B = ∠ E$ (1 mark)
(iii)(b) Concludes SAS similarity (½ mark)

Revise the chapter

Chapter: Triangles (notes, formulas, where marks are lost and every past question)
Topic: Basic Proportionality Theorem (Thales Theorem)

Practise this chapterBoard-style questions from Triangles, one at a time, marked against the answer key and marking points.Back to the full paperSample paper 2024-25 (Standard): every question, by section.

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CBSE Class 10 Mathematics Standard sample paper 2024-25: question 37

Hints

Step-by-step solution

Revise the chapter

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