In a $Δ\ ABC,$ $D$ and $E$ are points on the sides $AB$ and $AC$ respectively. For each of the following cases show that $DE\ ∥\ BC$:
$AD\ =\ 5.7\ cm$, $BD\ =\ 9.5\ cm$, $AE\ =\ 3.3\ cm$, and $EC\ =\ 5.5\ cm$.

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Given:

In a $Δ\ ABC,$ $D$ and $E$ are points on the sides $AB$ and $AC$ respectively.

$AD\ =\ 5.7\ cm$, $BD\ =\ 9.5\ cm$, $AE\ =\ 3.3\ cm$, and $EC\ =\ 5.5\ cm$.

To do:

We have to prove that $DE\ ∥\ BC$.

Solution:

We know that,

The Converse of the Basic proportionality theorem states that "If a line divides any of the two sides of a triangle in the same ratio, then that line is parallel to the third side". 

Here,

$\frac{AD}{DB}=\frac{5.7}{9.5}=\frac{3}{5}$

$\frac{AE}{EC}=\frac{3.3}{5.5}=\frac{3}{5}$

Therefore,

$\frac{AD}{DB}=\frac{AE}{EC}$  

Hence, by the converse of Basic proportionality theorem,

$DE\ ∥\ BC$.

Hence proved. 

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Updated on: 10-Oct-2022

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