Question : $D$ and $E$ are points on the sides $AB$ and $AC$ respectively of $ riangle ABC$ such that $DE$ is parallel to $BC$ and $AD: DB = 4:5$, $CD$ and $BE$ intersect each other at $F$. Find the ratio of the areas of $ riangle DEF$ and $ riangle CBF$.
Option 1: $16:25$
Option 2: $16:81$
Option 3: $81:16$
Option 4: $4:9$

Question

Question : $D$ and $E$ are points on the sides $AB$ and $AC$ respectively of $ riangle ABC$ such that $DE$ is parallel to $BC$ and $AD: DB = 4:5$, $CD$ and $BE$ intersect each other at $F$. Find the ratio of the areas of $ riangle DEF$ and $ riangle CBF$.

Option 1: $16:25$

Option 2: $16:81$

Option 3: $81:16$

Option 4: $4:9$

Team Careers360 · Accepted Answer

Correct Answer: $16:81$

Solution :
In $ riangle ADE$ and $ riangle ABC$
$\angle AED=\angle C$ (corresponding angle)
$\angle ADE=\angle B$ (corresponding angle)
$\angle A=\angle A$ (common angle)
$ riangle ADE\sim riangle ABC$ (by AAA criteria)
$\frac{AD}{AB}=\frac{AE}{AC}=\frac{DE}{BC}=\frac{4}{9}$
In $ riangle DEF$ and $ riangle BFC$
$\angle EDF=\angle BCF$ (alternate interior angle)
$\angle DEF=\angle CBF$ (alternate interior angle)
$ riangle DEF\sim riangle CBF$ (by AA criteria)
The theorem used is the two triangles are similar, and the ratio of areas is equal to the ratio of squares of corresponding sides.
$\frac{ar riangle DEF}{ar riangle BFC}=(\frac{DE}{BC})^2=(\frac{4}{9})^2=\frac{16}{81}$
Hence, the correct answer is $16:81$.

Exams

Colleges

Predictors

Resources

Quick links

Exams

Colleges & Courses

Predictors

Resources

Exams

Colleges

Predictors & E-Books

Resources

Engineering Preparation

Medical Preparation

Law Preparation

MBA Preparation

Exams

Colleges

Resources

Exams

Colleges

Predictors & Articles

Resources

Exams

Colleges

Predictors

Resources

Exams

Colleges

Upcoming Events

Resources

Exams

Colleges

Resources

Quick Links

Top Streams

Specializations

Resources

Top Providers

Exams

Colleges

Resources

Diploma Colleges

Exams

Colleges

Resources

Exams

Resources

Top Courses & Careers

Colleges

Exams

Upcoming Events

Resources

Other Exams

Exams

Colleges

Top Countries

Resources

Exams

Top Schools

Products & Resources

NCERT Study Material

Question : D and E are points on the sides AB and AC respectively of △ABC such that DE is parallel to BC and AD:DB=4:5, CD and BE intersect each other at F. Find the ratio of the areas of △DEF and △CBF.Option 1: 16:25Option 2: 16:81Option 3: 81:16Option 4: 4:9

Related Questions

Know More about

Upcoming Exams

Trending Articles around SSC CGL

Download the Careers360 App on your Android phone

Sign In/Sign Up