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Triangles
RD Sharma
CBSE
Grade 10

Question:

Prove that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding medians. (Note: Using similarity ratios). Use this to solve:
If $\Delta ABC \sim \Delta PQR$ with medians $AD$ and $PS$, and $AB = 6\text{ cm}, PQ = 9\text{ cm}, AD = 4\text{ cm}$, find $PS$.

Step-by-Step Solution

Key Concept: Part 1: $\Delta ABC \sim \Delta PQR \Rightarrow \Delta ABD \sim \Delta PQS \Rightarrow \dfrac{AB}{PQ} = \dfrac{AD}{PS}$. Part 2: $\dfrac{6}{9} = \dfrac{4}{PS} \Rightarrow \dfrac{2}{3} = \dfrac{4}{PS} \Rightarrow PS = 6\text{ cm}$.
Part 1: Proof that ratio of corresponding medians equals ratio of corresponding sides ($AB/PQ = AD/PS$). [2.5 Marks]
Part 2: Given $\Delta ABC \sim \Delta PQR$ with medians $AD, PS$. $\dfrac{AB}{PQ} = \dfrac{AD}{PS}$. [1.0 Mark]
$\dfrac{6}{9} = \dfrac{4}{PS} \Rightarrow \dfrac{2}{3} = \dfrac{4}{PS}$. [1.0 Mark]
$2 PS = 12 \Rightarrow PS = 6\text{ cm}$. [0.5 Mark]

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🎯 Official CBSE Marking Scheme:
Proving median ratio equals side ratio: 2.5 Marks
Setting ratio $\dfrac{6}{9} = \dfrac{4}{PS}$: 1.0 Mark
Solving $PS = 6\text{ cm}$: 1.5 Marks

Correct Answer:
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