MathsCBSE Class 10 · Standard · Sample paper, session 2026-27 · 5 marks · Long answer

CBSE Class 10 Mathematics Standard sample paper 2026-27: question 33

From the official CBSE sample question paper for 2026-27, section D, question 33, for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q33 · 5 marks · Long answerTriangles

Prove that a line drawn parallel to one side of a triangle to intersect the other two sides in distinct points, divides the other two sides in the same ratio.
Using the above theorem solve the following: PQRS is a trapezium with PQ $∥$ SR. X and Y are points on non-parallel sides PS and QR respectively such that XY $∥$ PQ. Show that $\frac{P X}{X S} = \frac{Q Y}{Y R}$ .

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2026-27.

Hints

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

Hint 1

For the theorem, compare areas of triangles that share a vertex at D or E. Join BE and CD and drop perpendiculars to use as heights.

Hint 2

Triangles on the same base and between the same parallels have equal areas; use this for $△ B D E$ and $△ D E C$ .

Hint 3

For the trapezium, draw the diagonal PR so that XY is split into two parts, each parallel to a side of a triangle. Then apply the theorem twice.

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.

Figure: Draw $△ A B C$ with a line DE parallel to BC, meeting AB at D and AC at E.
Given: In $△ A B C$ , $D E ∥ B C$ , with D on AB and E on AC.
To prove: $\frac{A D}{D B} = \frac{A E}{E C}$ .
Construction: Join BE and CD. Draw $E N ⊥ A B$ and $D M ⊥ A C$ .
1 mark
Proof: Use area $= \frac{1}{2} \times$ base $\times$ height, with $E N$ as the common height:
$\frac{a r ( △ A D E )}{a r ( △ B D E )} = \frac{\frac{1}{2} \times A D \times E N}{\frac{1}{2} \times D B \times E N} = \frac{A D}{D B}$ …(1)
½ mark
Similarly, with $D M$ as the common height:
$\frac{a r ( △ A D E )}{a r ( △ D E C )} = \frac{\frac{1}{2} \times A E \times D M}{\frac{1}{2} \times E C \times D M} = \frac{A E}{E C}$ …(2)
½ mark
$△ B D E$ and $△ D E C$ are on the same base DE and between the same parallels DE and BC.
So $a r (△ B D E) = a r (△ D E C)$ …(3)
½ mark
From (1), (2) and (3): $\frac{A D}{D B} = \frac{A E}{E C}$ . Hence the theorem is proved.
½ mark
Application.
Figure: Draw trapezium PQRS with $P Q ∥ S R$ , X on PS and Y on QR, with $X Y ∥ P Q$ .
Join the diagonal PR. Let it cut XY at M.
Since $X Y ∥ P Q$ and $P Q ∥ S R$ , we get $X Y ∥ S R$ , so $X M ∥ S R$ .
½ mark
In $△ P S R$ , $X M ∥ S R$ . By the theorem above (BPT):
$\frac{P X}{X S} = \frac{P M}{M R}$ …(i)
½ mark
In $△ R P Q$ , $M Y ∥ P Q$ . By BPT:
$\frac{R M}{M P} = \frac{R Y}{Y Q}$ , that is, $\frac{P M}{M R} = \frac{Q Y}{Y R}$ …(ii)
½ mark
From (i) and (ii): $\frac{P X}{X S} = \frac{Q Y}{Y R}$ . Hence shown.
½ mark

Answer: Theorem (BPT) proved; using it, $\frac{P X}{X S} = \frac{Q Y}{Y R}$ is shown.

Where the marks are

Correct figure, given, to prove and construction for the theorem (1 mark)
Writes area ratios of $△ A D E$ to $△ B D E$ and $△ D E C$ as $\frac{A D}{D B}$ and $\frac{A E}{E C}$ (1 mark)
Uses equal areas of $△ B D E$ and $△ D E C$ (same base, same parallels) and concludes the theorem (1 mark)
Joins PR meeting XY at M (½ mark)
Applies BPT in $△ P S R$ to get $\frac{P X}{X S} = \frac{P M}{M R}$ (½ mark)
Applies BPT in $△ P QR$ to get $\frac{Q Y}{Y R} = \frac{P M}{M R}$ (½ mark)
Concludes $\frac{P X}{X S} = \frac{Q Y}{Y R}$ (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 2026-27 (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.

CBSE Class 10 Mathematics Standard sample paper 2026-27: question 33

Hints

Step-by-step solution

Revise the chapter

More questions from Triangles