MathsCBSE Class 10 · Basic · Sample paper, session 2025-26 · 5 marks · Long answer

CBSE Class 10 Mathematics Basic sample paper 2025-26: question 32

From the official CBSE sample question paper for 2025-26, section D, question 32, for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q32 · 5 marks · Long answerTriangles

Prove that a line drawn parallel to one side of a triangle intersecting other two sides in distinct points, divides the other two sides in the same ratio.

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2025-26.

Hints

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

Hint 1

This is the Basic Proportionality (Thales) Theorem; the proof uses areas of triangles.

Hint 2

Join BE and CD and draw perpendiculars from E to AB and from D to AC to use as heights.

Hint 3

Triangles BDE and CED stand on the same base DE between parallels DE and BC.

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.

Given: In $△ A B C$ , a line $l ∥ B C$ meets AB at D and AC at E.
½ mark
To prove: $\frac{A D}{D B} = \frac{A E}{E C}$
½ mark
Construction: Join BE and CD. Draw $D M ⊥ A C$ and $E N ⊥ A B$ .
½ mark
Figure: Triangle ABC with D on AB, E on AC, DE ∥ BC; dashed segments BE and CD; N the foot of perpendicular from E on AB, M the foot of perpendicular from D on AC.
½ mark
Proof: $\frac{a r ( △ A D E )}{a r ( △ B D E )} = \frac{\frac{1}{2} \cdot A D \cdot E N}{\frac{1}{2} \cdot D B \cdot E N} = \frac{A D}{D B}$ ...(1)
1 mark
$\frac{a r ( △ A D E )}{a r ( △ C E D )} = \frac{\frac{1}{2} \cdot A E \cdot D M}{\frac{1}{2} \cdot E C \cdot D M} = \frac{A E}{E C}$ ...(2)
½ mark
$a r (△ B D E) = a r (△ C E D)$ ...(3) [triangles on the same base DE and between the same parallels DE and BC]
½ mark
From (1), (2) and (3): $\frac{a r ( △ A D E )}{a r ( △ B D E )} = \frac{a r ( △ A D E )}{a r ( △ C E D )} \Rightarrow \frac{A D}{D B} = \frac{A E}{E C}$ . Hence proved.
1 mark

Answer: $\frac{A D}{D B} = \frac{A E}{E C}$ (Basic Proportionality Theorem proved)

Where the marks are

Given (½ mark)
To prove (½ mark)
Construction (½ mark)
Correct figure (½ mark)
Area ratio giving $\frac{A D}{D B}$ (1 mark)
Area ratio giving $\frac{A E}{E C}$ (½ mark)
Equal areas of $△ B D E$ and $△ C E D$ with reason (½ mark)
Combining to conclude (1 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 2025-26 (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.

CBSE Class 10 Mathematics Basic sample paper 2025-26: question 32

Hints

Step-by-step solution

Revise the chapter

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