In fig. MN || BC and AM : MB = 1 : 2, then ar( ) ar( ) = ________… - Vidyadip

Question

In fig. MN || BC and AM : MB = 1 : 2, then $\frac{\text{ar}(\triangle\text{AMN})}{\text{ar}(\triangle\text{ABC})}=$ _________.

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Answer

In fig. MN || BC and AM : MB = 1 : 2, then $\frac{\text{ar}(\triangle\text{AMN})}{\text{ar}(\triangle\text{ABC})}=\frac{1}{9}$ $$ Solution: Since we have given that AM : MB = 1 : 2 So, AB = 1 + 2 = 3 Since MN || BC So, Using "Area similarity theorem" ,we get that$\frac{\text{ar}(\triangle\text{AMN})}{\text{ar}(\triangle\text{ABC})}=\frac{\text{AM}^2}{\text{AB}^2}=\frac{1}{3^2}=\frac{1}{9}$
Hence, the required ratio is 1 : 9.

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