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In the adjoining figure, $IE \parallel TR$ and $\frac{GI}{IT} = \frac{1}{2}$. Find $\frac{TR}{IE}$.
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Ans. As $IE \parallel TR$
$\angle GIE = \angle GTR\angle GEI = \angle GRT\}$ (corresponding angles) (1 Mark)
$\triangle GIE \sim \triangle GTR$ (by AA Similarity) (1 Mark)
$\frac{GI}{GT} = \frac{GE}{GR} = \frac{IE}{TR}$ --- (1) (1 Mark)
Using $\frac{GI}{IT} = \frac{1}{2}$ to get
$\frac{GT}{GI} = \frac{3}{1}$ --- (2) (1 Mark)
Using (1) and (2) to get
$\frac{TR}{IE} = \frac{3}{1}$ (1 Mark)
$\angle GIE = \angle GTR\angle GEI = \angle GRT\}$ (corresponding angles) (1 Mark)
$\triangle GIE \sim \triangle GTR$ (by AA Similarity) (1 Mark)
$\frac{GI}{GT} = \frac{GE}{GR} = \frac{IE}{TR}$ --- (1) (1 Mark)
Using $\frac{GI}{IT} = \frac{1}{2}$ to get
$\frac{GT}{GI} = \frac{3}{1}$ --- (2) (1 Mark)
Using (1) and (2) to get
$\frac{TR}{IE} = \frac{3}{1}$ (1 Mark)