- Data Structure
- Networking
- RDBMS
- Operating System
- Java
- MS Excel
- iOS
- HTML
- CSS
- Android
- Python
- C Programming
- C++
- C#
- MongoDB
- MySQL
- Javascript
- PHP
- Physics
- Chemistry
- Biology
- Mathematics
- English
- Economics
- Psychology
- Social Studies
- Fashion Studies
- Legal Studies
- Selected Reading
- UPSC IAS Exams Notes
- Developer's Best Practices
- Questions and Answers
- Effective Resume Writing
- HR Interview Questions
- Computer Glossary
- Who is Who
In figure below, $D$ and $ \mathrm{E} $ are two points on $ \mathrm{BC} $ such that $ \mathrm{BD}=\mathrm{DE}=\mathrm{EC} $. Show that $ \operatorname{ar}(\mathrm{ABD})=\operatorname{ar}(\mathrm{ADE})=\operatorname{ar}(\mathrm{AEC}) $.
"
Given:
$D$ and \( \mathrm{E} \) are two points on \( \mathrm{BC} \) such that \( \mathrm{BD}=\mathrm{DE}=\mathrm{EC} \).
To do:
We have to show that \( \operatorname{ar}(\mathrm{ABD})=\operatorname{ar}(\mathrm{ADE})=\operatorname{ar}(\mathrm{AEC}) \).
Solution:
In $\triangle ABE$
$BD=DE$
This implies,
$AD$ is the median.
We know that,
The median of a triangle divides it into two parts of equal areas.
This implies,
$ar(\triangle ABD) = ar(\triangle AED)$.........(i)
In $\triangle ADC$,
$DE=EC$
$AE$ is the median
This implies,
$ar(\triangle ADE) = ar(\triangle AEC)$..........(ii)
From (i) and (ii), we get,
$ar(\triangle ABD) = ar(\triangle ADE) = ar(\triangle AEC)$
Hence proved.
Advertisements
To Continue Learning Please Login
Login with Google