In $Δ A B C, D$ and E are the midpoints of AB and AC respectively. Find the ratio of the areas of $Δ A D E a n d Δ A B C$ .

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Solution

The line joining the midpoints of two sides of a triangle is parallel to the third side. So, DE || BC.

Now, in $△ A D E$ and $△ A B C$ , $∠ A = ∠ A$

and $∠ A D E = ∠ A B C$ (Corresponding angles)

Hence, ΔABC ~ ΔADE [By AA similarity theorem]
$t h e n fraction numerator a r e a space o f space increment A B C over denominator a r e a space o f increment A D E end fraction equals fraction numerator A B squared over denominator A D squared end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals open parentheses 2 over 1 close parentheses squared fraction numerator a r e a space o f space increment A B C over denominator a r e a space o f increment A D E end fraction equals 4$

Hence area of ΔADE : area of ΔABC = 1 : 4

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In ΔABC,D and E are the midpoints of AB and AC respectively. Find the ratio of the areas of ΔADE and ΔABC.

The line joining the midpoints of two sides of a triangle is parallel to the third side. So, DE || BC. Now, in △ADE and △ABC, ∠A=∠A and ∠ADE=∠ABC (Corresponding angles) Hence, ΔABC ~ ΔADE [By AA similarity theorem] Hence area of ΔADE : area of ΔABC = 1 : 4

In $Δ A B C, D$ and E are the midpoints of AB and AC respectively. Find the ratio of the areas of $Δ A D E a n d Δ A B C$ .