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

CBSE Class 10 Mathematics Basic sample paper 2026-27: question 32

From the official CBSE sample question paper for 2026-27, 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 if a line is drawn parallel to one side of a triangle intersecting the other two sides in distinct points, then the other two sides are divided in the same ratio.

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

Join BE and CD and drop perpendiculars to compare the areas of triangles with the same height.

Hint 2

Triangles BDE and DEC lie on the same base DE and between the same parallels, so their areas are equal.

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. Join BE and CD, and draw $E N ⊥ A B$ and $D M ⊥ A C$ .
½ mark
Given: In $△ A B C$ , $D E ∥ B C$ , and DE 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 $E N ⊥ A B$ and $D M ⊥ A C$ .
½ mark
Proof: $a r (A D E) = \frac{1}{2} \times A D \times E N$ and $a r (B D E) = \frac{1}{2} \times D B \times E N$ , so $\frac{a r ( A D E )}{a r ( B D E )} = \frac{A D}{D B}$ ...(1)
1 mark
Similarly, $a r (A D E) = \frac{1}{2} \times A E \times D M$ and $a r (D E C) = \frac{1}{2} \times E C \times D M$ , so $\frac{a r ( A D E )}{a r ( D E C )} = \frac{A E}{E C}$ ...(2)
1 mark
$△ B D E$ and $△ D E C$ lie 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)
From (1), (2) and (3): $\frac{A D}{D B} = \frac{A E}{E C}$ . Hence proved.
1 mark

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

Where the marks are

Correct figure (½ mark)
Given (½ mark)
To prove (½ mark)
Construction (½ mark)
Ratio of areas ADE : BDE = AD : DB (1 mark)
Ratio of areas ADE : DEC = AE : EC (1 mark)
Equal areas of BDE and DEC (same base, same parallels) and conclusion (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 2026-27 (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 2026-27: question 32

Hints

Step-by-step solution

Revise the chapter

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