Question

∆DEF ~ ∆ABC ; If DE : AB = 2 : 3 and ar(∆DEF) is equal to 44 square units, then find ar(∆ABC) in square units. ​

Answer 1

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

↝ ∆DEF ~ ∆ABC

↝ DE : AB = 2 : 3

↝ ar(∆DEF) is equal to 44 square units

$\large\underline{\sf{To\:Find - }}$

↝ ar(∆ABC)

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

Given that,

∆DEF ~ ∆ABC

DE : AB = 2 : 3

and ar(∆DEF) is equal to 44 square units

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 Area Ratio Theorem, we have

$\rm :\longmapsto\:\dfrac{ar( \triangle \: ABC)}{ar(\triangle \: DEF)} = \dfrac{ {AB}^{2} }{ {DE}^{2} }$

On Substituting the given values, we get

$\rm :\longmapsto\:\dfrac{ar( \triangle \: ABC)}{44} = \dfrac{ {3}^{2} }{ {2}^{2} }$

$\rm :\longmapsto\:\dfrac{ar( \triangle \: ABC)}{44} = \dfrac{9}{4}$

$\rm :\longmapsto\:\dfrac{ar( \triangle \: ABC)}{11} = \dfrac{9}{1}$

Hence, ar(∆ABC) = 99 square units

Additional Information :-

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. 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.