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Triangles
CBSE 2026 Board Exam Set 1 (Code 30/7/1)
CBSE_BOARD_PYQ_2026_30_7_1
Grade 10
Question:
[Section D]
State and prove Basic Proportionality Theorem.
OR
In a trapezium $ABCD$ with $AB \parallel DC$, a line segment $EF$ is drawn parallel to $AB$ intersecting $AD$ at $E$ and $BC$ at $F$. Prove that $\dfrac{AE}{ED} = \dfrac{BF}{FC}$.
State and prove Basic Proportionality Theorem.
OR
In a trapezium $ABCD$ with $AB \parallel DC$, a line segment $EF$ is drawn parallel to $AB$ intersecting $AD$ at $E$ and $BC$ at $F$. Prove that $\dfrac{AE}{ED} = \dfrac{BF}{FC}$.
Step-by-Step Solution
Key Concept: Main: BPT proof using triangle area ratios. OR: Join diagonal $AC$ intersecting $EF$ at $G$, apply BPT twice.
[Main Question Solution]
Statement, Given, To Prove, Construction and complete area ratio proof. [5.0 Marks]
-------------------------------
[OR Choice Question Solution]
Join $AC$ intersecting $EF$ at $G$. In $\triangle CAB$, $FG \parallel AB \Rightarrow \dfrac{CF}{FB} = \dfrac{CG}{GA}$. Eq (1). [2.0 Marks]
In $\triangle ADC$, $EG \parallel DC \Rightarrow \dfrac{CG}{GA} = \dfrac{DE}{EA}$. Eq (2). [2.0 Marks]
From (1) and (2), $\dfrac{DE}{EA} = \dfrac{CF}{FB} \Rightarrow \dfrac{AE}{ED} = \dfrac{BF}{FC}$. Hence proved. [1.0 Mark]
Statement, Given, To Prove, Construction and complete area ratio proof. [5.0 Marks]
-------------------------------
[OR Choice Question Solution]
Join $AC$ intersecting $EF$ at $G$. In $\triangle CAB$, $FG \parallel AB \Rightarrow \dfrac{CF}{FB} = \dfrac{CG}{GA}$. Eq (1). [2.0 Marks]
In $\triangle ADC$, $EG \parallel DC \Rightarrow \dfrac{CG}{GA} = \dfrac{DE}{EA}$. Eq (2). [2.0 Marks]
From (1) and (2), $\dfrac{DE}{EA} = \dfrac{CF}{FB} \Rightarrow \dfrac{AE}{ED} = \dfrac{BF}{FC}$. Hence proved. [1.0 Mark]
Correct Answer: Proof of BPT or Trapezium parallel segment ratio.
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