MathsCBSE Class 10 · Standard · Board exam 2025, set 30/2/1 · 5 marks · Long answer

CBSE Class 10 Mathematics Standard 2025: question 34

Set in the CBSE Class 10 board exam of 2025, paper 30/2/1, section D, question 34, for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q34 · 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. Hence, in the figure given below, prove that $\frac{A M}{M B} = \frac{A N}{N D}$ where $L M ∥ C B$ and $L N ∥ C D$ .

Figure in the paper: Points A (left), B (top), C (right) and D (bottom). Line AC is drawn. M lies on segment AB and N lies on segment AD; L lies on AC. Segments LM (parallel to CB) and LN (parallel to CD) are drawn. See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2025 board paper, set 30/2/1.

Hints

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

Hint 1

For the theorem, compare areas of triangles that share the same height. Join each end of the base to the opposite point on the parallel line.

Hint 2

Triangles on the same base and between the same parallels have equal areas.

Hint 3

For the figure, $L$ lies on $A C$ , which is common to $△ A B C$ and $△ A D C$ . Use the BPT in each triangle and compare both ratios with $\frac{A L}{L C}$ .

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.

Theorem (Basic Proportionality Theorem).
Given: In $△ P QR$ , a line $S T ∥ QR$ meets $P Q$ at $S$ and $P R$ at $T$ .
To prove: $\frac{P S}{S Q} = \frac{P T}{T R}$
Construction: Join $QT$ and $R S$ . Draw $T K ⊥ P Q$ and $S H ⊥ P R$ .
Figure: $△ P QR$ with $P$ at the top and $QR$ as the base. Segment $S T$ is drawn parallel to $QR$ , with $S$ on $P Q$ and $T$ on $P R$ . Segments $QT$ and $R S$ are joined, and the perpendiculars $T K$ (onto $P Q$ ) and $S H$ (onto $P R$ ) are shown.
(Letters $P, Q, R$ are used here so they do not clash with the letters of the second part.)
1½ marks
Proof:
$ar (P S T) = \frac{1}{2} \cdot P S \cdot T K$ and $ar (QS T) = \frac{1}{2} \cdot S Q \cdot T K$ , so
$\frac{ar ( P S T )}{ar ( QS T )} = \frac{P S}{S Q} ... (1)$
$ar (P S T) = \frac{1}{2} \cdot P T \cdot S H$ and $ar (R T S) = \frac{1}{2} \cdot T R \cdot S H$ , so
$\frac{ar ( P S T )}{ar ( R T S )} = \frac{P T}{T R} ... (2)$
$△ QS T$ and $△ R T S$ have the same base $S T$ and lie between the same parallels $S T$ and $QR$ . Hence $ar (QS T) = ar (R T S)$ ...(3)
From (1), (2) and (3): $\frac{P S}{S Q} = \frac{P T}{T R}$ . Hence proved.
1½ marks
Application. In $△ A B C$ , $L M ∥ C B$ , with $M$ on $A B$ and $L$ on $A C$ .
By the BPT, $\frac{A M}{M B} = \frac{A L}{L C}$ ...(4)
1 mark
In $△ A D C$ , $L N ∥ C D$ , with $N$ on $A D$ and $L$ on $A C$ .
By the BPT, $\frac{A N}{N D} = \frac{A L}{L C}$ ...(5)
½ mark
From (4) and (5), $\frac{A M}{M B} = \frac{A N}{N D}$ . Hence proved.
½ mark

Answer: BPT proved by the area method; then $\frac{A M}{M B} = \frac{A L}{L C} = \frac{A N}{N D}$ , so $\frac{A M}{M B} = \frac{A N}{N D}$

Where the marks are

Correct figure with given, to prove and construction (1½ marks)
Correct area-ratio proof of the BPT, using equal areas of triangles on the same base between the same parallels (1½ marks)
Applies the BPT in $△ A B C$ to get $\frac{A M}{M B} = \frac{A L}{L C}$ (1 mark)
Applies the BPT in $△ A D C$ to get $\frac{A N}{N D} = \frac{A L}{L C}$ (½ mark)
Concludes $\frac{A M}{M B} = \frac{A N}{N D}$ (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 paper2025 board paper, set 30/2/1 (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 2025: question 34

Hints

Step-by-step solution

Revise the chapter

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