- 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
Sides AB and BC and median AD of a triangle ABC are respectively proportional to sides PQ and QR and median PM of $∆PQR$ (see in given figure). Show that $∆ABC \sim ∆PQR$.
Given:
Sides AB and BC and median AD of a triangle ABC are respectively proportional to sides PQ and QR and median PM of $∆PQR$.
To do:
We have to show that $∆ABC \sim ∆PQR$.
Solution:
In $\triangle ABC$ and $\triangle PQR$,
$\frac{\mathrm{AB}}{\mathrm{PQ}}=\frac{\mathrm{BC}}{\mathrm{QR}}=\frac{\mathrm{AD}}{\mathrm{PM}}$
$\frac{\mathrm{AB}}{\mathrm{PQ}}=\frac{\frac{1}{2} \mathrm{BC}}{\frac{1}{2} \mathrm{QR}}=\frac{\mathrm{AD}}{\mathrm{PM}}$
$\frac{\mathrm{AB}}{\mathrm{PQ}}=\frac{\mathrm{BD}}{\mathrm{QM}}=\frac{\mathrm{AD}}{\mathrm{PM}}$
Therefore, by SSS criterion,
$\triangle \mathrm{ABD} \sim \triangle \mathrm{PQM}$
This implies,
$\angle B=\angle Q$ (CPCT)
In $\triangle \mathrm{ABC} \sim \triangle \mathrm{PQR}$,
$\angle B=\angle Q$
$\frac{\mathrm{AB}}{\mathrm{PQ}}=\frac{\mathrm{BC}}{\mathrm{QR}}$
Therefore, by SAS criterion,
$\triangle \mathrm{ABC} \sim \triangle \mathrm{PQR}$