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Triangles
RD Sharma
CBSE
Grade 10
Question:
State and Prove Basic Proportionality Theorem (BPT / Thales Theorem).
Step-by-Step Solution
Key Concept: Given: $\Delta ABC$ with $DE \parallel BC$. To Prove: $\dfrac{AD}{DB} = \dfrac{AE}{EC}$. Construction: Join $BE, CD$. Draw $EN \perp AB, DM \perp AC$. Proof via area ratios.
Statement: If a line is drawn parallel to one side of a triangle, it divides the other two sides in the same ratio. [0.5 Mark]
Construction: Join $BE$ and $CD$. Draw $EN \perp AB$ and $DM \perp AC$. [0.5 Mark]
$\text{Area}(\Delta ADE) = \dfrac{1}{2} \times AD \times EN$. $\text{Area}(\Delta BDE) = \dfrac{1}{2} \times DB \times EN \Rightarrow \dfrac{\text{Area}(ADE)}{\text{Area}(BDE)} = \dfrac{AD}{DB}$. (1) [1.0 Mark]
$\text{Area}(\Delta ADE) = \dfrac{1}{2} \times AE \times DM$. $\text{Area}(\Delta DEC) = \dfrac{1}{2} \times EC \times DM \Rightarrow \dfrac{\text{Area}(ADE)}{\text{Area}(DEC)} = \dfrac{AE}{EC}$. (2) [0.5 Mark]
Since $\Delta BDE$ and $\Delta DEC$ are on same base $DE$ and between same parallels $DE \parallel BC$, $\text{Area}(BDE) = \text{Area}(DEC)$. From (1) & (2): $\dfrac{AD}{DB} = \dfrac{AE}{EC}$. Proved! [0.5 Mark]
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🎯 Official CBSE Marking Scheme:
Statement & Construction: 1.0 Mark
Deriving area ratios (1) & (2): 1.5 Marks
Equating areas on same base $DE$ and concluding $\dfrac{AD}{DB} = \dfrac{AE}{EC}$: 0.5 Mark
Construction: Join $BE$ and $CD$. Draw $EN \perp AB$ and $DM \perp AC$. [0.5 Mark]
$\text{Area}(\Delta ADE) = \dfrac{1}{2} \times AD \times EN$. $\text{Area}(\Delta BDE) = \dfrac{1}{2} \times DB \times EN \Rightarrow \dfrac{\text{Area}(ADE)}{\text{Area}(BDE)} = \dfrac{AD}{DB}$. (1) [1.0 Mark]
$\text{Area}(\Delta ADE) = \dfrac{1}{2} \times AE \times DM$. $\text{Area}(\Delta DEC) = \dfrac{1}{2} \times EC \times DM \Rightarrow \dfrac{\text{Area}(ADE)}{\text{Area}(DEC)} = \dfrac{AE}{EC}$. (2) [0.5 Mark]
Since $\Delta BDE$ and $\Delta DEC$ are on same base $DE$ and between same parallels $DE \parallel BC$, $\text{Area}(BDE) = \text{Area}(DEC)$. From (1) & (2): $\dfrac{AD}{DB} = \dfrac{AE}{EC}$. Proved! [0.5 Mark]
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🎯 Official CBSE Marking Scheme:
Statement & Construction: 1.0 Mark
Deriving area ratios (1) & (2): 1.5 Marks
Equating areas on same base $DE$ and concluding $\dfrac{AD}{DB} = \dfrac{AE}{EC}$: 0.5 Mark
Correct Answer:
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