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In Fig. 6.33, $CM$ and $RN$ are medians of $\triangle ABC$ and $\triangle PQR$ respectively, and $\triangle ABC \sim \triangle PQR$. Which statement is correct? 
A$\triangle AMC \sim \triangle PNR$ by SAS similarity
B$\triangle AMC \cong \triangle PNR$ by SAS congruence
C$\triangle CMB \sim \triangle RNQ$ by AA similarity only
D$\frac{CM}{RN}=\frac{BC}{PQ}$
Answer & Solution
Correct answer: A. $\triangle AMC \sim \triangle PNR$ by SAS similarity
Since $\triangle ABC \sim \triangle PQR$, we have $\frac{AB}{PQ}=\frac{CA}{PR}$ and $\angle A=\angle P$. As $CM$ and $RN$ are medians, $AB=2AM$ and $PQ=2PN$, so $\frac{AM}{PN}=\frac{CA}{PR}$. With the included angle $\angle MAC=\angle NPR$, this gives $\triangle AMC \sim \triangle PNR$ by SAS. Option D is incorrect because the result proved is $\frac{CM}{RN}=\frac{AB}{PQ}$, not $\frac{BC}{PQ}$.
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