Question : In $ riangle ABC$, $D$ and $E$ are the points of sides $AB$ and $BC$ respectively such that $DE \parallel AC$ and $AD : DB = 3 : 2$. The ratio of the area of trapezium $ACED$ to that of $ riangle DBE$ is:
Option 1: $4:15$
Option 2: $15:4$
Option 3: $4:21$
Option 4: $21:4$

Question

Question : In $ riangle ABC$, $D$ and $E$ are the points of sides $AB$ and $BC$ respectively such that $DE \parallel AC$ and $AD : DB = 3 : 2$. The ratio of the area of trapezium $ACED$ to that of $ riangle DBE$ is:

Option 1: $4:15$

Option 2: $15:4$

Option 3: $4:21$

Option 4: $21:4$

Team Careers360 · Accepted Answer

Correct Answer: $21:4$

Solution :

Given that in $ riangle ABC$, $D$ and $E$ are the points of sides $AB$ and $BC$, $DE \parallel AC$ and $AD : DB = 3 : 2$.
From the figure,
$\frac{DB}{AB}=\frac{2}{5}$
In $ riangle ABC$ and $ riangle DBE$,
$\angle ABC=\angle DBE$ (common angle)
$\angle BAC=\angle BDE$ (corresponding angle)
$\angle ACB=\angle DEB$ (corresponding angle)
$ riangle ABC\sim riangle DBE$, by AAA similarity.
By Thales theorem, the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.
$ herefore\frac{ ext{area of $ riangle ABC$}}{ ext{area of $ riangle DBE$}}=(\frac{AB}{DB})^2$
$⇒\frac{ ext{area of $ riangle ABC$}}{ ext{area of $ riangle DBE$}}=(\frac{5}{2})^2$
$⇒\frac{ ext{area of $ riangle ABC$}}{ ext{area of $ riangle DBE$}}=\frac{25}{4}$
$⇒\frac{ ext{area of $ riangle ABC$}}{ ext{area of $ riangle DBE$}}-1 =\frac{25}{4}-1$
$⇒\frac{ ext{area of $ riangle ABC$}- ext{area of $ riangle DBE$}}{ ext{area of $ riangle DBE$}} =\frac{25-4}{4}$
$⇒\frac{ ext{area of trapezium $ACED$ }}{ ext{area of $ riangle DBE$}} =\frac{21}{4}$
Hence, the correct answer is $21:4$.

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Question : In △ABC, D and E are the points of sides AB and BC respectively such that DE∥AC and AD:DB=3:2. The ratio of the area of trapezium ACED to that of △DBE is:Option 1: 4:15Option 2: 15:4Option 3: 4:21Option 4: 21:4

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Question : In $△ A B C$ , $D$ and $E$ are the points of sides $A B$ and $B C$ respectively such that $D E ∥ A C$ and $A D : D B = 3 : 2$ . The ratio of the area of trapezium $A C E D$ to that of $△ D B E$ is:
Option 1: $4 : 15$
Option 2: $15 : 4$
Option 3: $4 : 21$
Option 4: $21 : 4$