If two triangles $\Delta ABC$ and $\Delta PQR$ are similar, prove that the ratio of their perimeters is equal to the ratio of their corresponding sides.
Step-by-Step Solution
Key Concept: $\Delta ABC \sim \Delta PQR \Rightarrow \dfrac{AB}{PQ} = \dfrac{BC}{QR} = \dfrac{CA}{RP} = k \Rightarrow AB = k PQ, BC = k QR, CA = k RP$.
Let $\dfrac{AB}{PQ} = \dfrac{BC}{QR} = \dfrac{CA}{RP} = k$. [1.0 Mark]
$AB = k PQ, BC = k QR, CA = k RP$. [1.0 Mark]
$\dfrac{\text{Perimeter}(\Delta ABC)}{\text{Perimeter}(\Delta PQR)} = \dfrac{AB + BC + CA}{PQ + QR + RP} = \dfrac{k(PQ + QR + RP)}{PQ + QR + RP} = k = \dfrac{AB}{PQ}$. Proved! [1.0 Mark]
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🎯 Official CBSE Marking Scheme:
Expressing sides as $k \times$ corresponding side: 1.5 Marks
Evaluating perimeter ratio $= k$: 1.5 Marks
Correct Answer: