Question

In the given figure, DE||BC and AE:EC=3:4. What will be the ratio of the areas AADE and AABC?​

Answer 1

$\large\underline{\sf{Solution-}}$

Given that,

In triangle ABC, DE || BC, such that AE : EC = 3 : 4

Let AE = 3x and EC = 4x, so it means AC = 7x

Now, In triangle ADE and triangle ABC

$\rm :\longmapsto\: \angle ADE\: = \: \angle ABC \: \: \: \: \{corresponding \: angles \}$

$\rm :\longmapsto\: \angle AED\: = \: \angle ACB \: \: \: \: \{corresponding \: angles \}$

$\rm \implies\:\triangle AED \: \sim \: \triangle ACB \: \: \: \: \{AA \: Similarity \}$

We know,

Area Ratio Theorem :- This theorem states that, the ratio of the area of two similar triangles is equal to the ratio of the squares of corresponding sides.

So, using this

$\rm :\longmapsto\:\dfrac{ar(\triangle ADE)}{ar(\triangle ABC)} = \dfrac{ {AE}^{2} }{ {AC}^{2} }$

$\rm :\longmapsto\:\dfrac{ar(\triangle ADE)}{ar(\triangle ABC)} = \dfrac{ {(3x)}^{2} }{ {(7x)}^{2} }$

$\rm :\longmapsto\:\dfrac{ar(\triangle ADE)}{ar(\triangle ABC)} = \dfrac{9}{ 49}$

Therefore,

$\bf\implies \:ar(\triangle ADE) : ar(\triangle ABC) \: = \: 9 : 49$

• So, option (d) is correct.

Explore more :-

1. Pythagoras Theorem :-

This theorem states that : In a right-angled triangle, the square of the longest side is equal to sum of the squares of remaining sides.

2. Converse of Pythagoras Theorem :-

This theorem states that : If the square of the longest side is equal to sum of the squares of remaining two sides, angle opposite to longest side is right angle.

3. Area Ratio Theorem :-

This theorem states that :- The ratio of the area of two similar triangles is equal to the ratio of the squares of corresponding sides.

4. Basic Proportionality Theorem

If a line is drawn parallel to one side of a triangle, intersects the other two lines in distinct points, then the other two sides are divided in the same ratio.

Answer 2

Answer:

Option (d) is correct 9 : 49