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Home » In ∆ABC, D and E are the midpoints of AB and AC respectively. Find the ratio of the areas of ∆ADE and ∆ABC.
In ∆ABC, D and E are the midpoints of AB and AC respectively. Find the ratio of the areas of ∆ADE and ∆ABC.
CBSE | Class 10 | Exercise 4C | Maths | RS Aggarwal | Triangles
In ∆ABC, D and E are the midpoints of AB and AC respectively. Find the ratio of the areas of ∆ADE and ∆ABC.

 

 

 

 

Answer:

Given,

D and E are midpoints of AB and AC.

Applying midpoint theorem,

DE ‖ BC.

By B.P.T.,

\frac{A D}{A B}=\frac{A E}{A C}

∠???? = ∠????

Applying SAS similarity theorem,

∆ ADE~ ∆ ABC.

The ratio of areas of these triangles will be equal to the ratio of squares of their corresponding sides.

\begin{aligned} \frac{\operatorname{ar}(\triangle A D E)}{a r(\triangle A B C)} &=\frac{D E^{2}}{B C^{2}} \\ & \Rightarrow \frac{\left(\frac{1}{2} B C^{2}\right)}{B C^{2}} \\ & \Rightarrow \frac{1}{4} \end{aligned}

 

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