- Trending Categories
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 fig. 3, ABC is a right triangle, right angled at C and D is the mid-point of BC, Prove that $( AB)^{2} =4( AD)^{2} -3( AC)^{2} .$
"
Given: ABC is a right triangle, right angled at C and D is the mid-point of BC.
To do: To Prove that $( AB)^{2} =4( AD)^{2} -3( AC)^{2} .$
Solution:
Here given $AC\bot BC$ $( \because \vartriangle ABC\ is\ a\ right\ triangle)$
$BD=CD$ $( \because \ D\ is\ the\ mid-point\ of\ BC)$
In $\vartriangle$ABC,
$( Using Pythagoras\ theorem)$
$( AB)^{2} =( BC)^{2} +( AC)^{2}$
$\Rightarrow ( AB)^{2} =( BD+CD)^{2} +( AC)^{2}$ $( \because \ BD+CD=BC)$
$\Rightarrow ( AB)^{2} =( CD+CD)^{2} +( AC)^{2}$
$\Rightarrow ( AB)^{2} =4( CD)^{2} +( AC)^{2}$ $\dotsc \dotsc \dotsc \dotsc \dotsc \dotsc \dotsc ( 1)$
In triangle $\vartriangle ADC$
$( On\ using\ pythagoras\ theorem)$
$( AD)^{2} =( AC)^{2} +( CD)^{2}$ $( \because \vartriangle ADC\ is\ also\ a\ right\ triangle)$
$\Rightarrow ( CD)^{2} =( AD)^{2} -( AC)^{2}$
$After\ substituting\ the\ value\ of\ ( CD)^{2} \ in\ equation\ ( 1)$
$\Rightarrow ( AB)^{2} =4\left(( AD)^{2} -( AC)^{2}\right) +( AC)^{2}$
$\Rightarrow ( AB)^{2} =4( AD)^{2} -4( AC)^{2} +( AC)^{2}$
$\Rightarrow ( AB)^{2} =4( AD)^{2} -3( AC)^{2}$
Hence proved.
Updated on: 10-Oct-2022
34 Views
Advertisements