- 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
Show that $A (-3, 2), B (-5, -5), C (2, -3)$ and $D (4, 4)$ are the vertices of a rhombus.
Given:
Given points are $(-3, 2), (-5, -5), (2, -3)$ and $(4, 4)$.
To do:
We have to show that the $(-3, 2), (-5, -5), (2, -3)$ and $(4, 4)$ are the vertices of a rhombus.
Solution:
Let \( \mathrm{ABCD} \) is a quadrilateral whose vertices are \( \mathrm{A}(-3,2), \mathrm{B}(-5,-5), \mathrm{C}(2,-3) \) and \( \mathrm{D}(4,4) \).
We know that,
The distance between two points \( \mathrm{A}\left(x_{1}, y_{1}\right) \) and \( \mathrm{B}\left(x_{2}, y_{2}\right) \) is \( \sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}} \).
Therefore,
\( A B=\sqrt{(-5+3)^{2}+(-5-2)^{2}} \)\( =\sqrt{(-2)^{2}+(-7)^{2}} \)
\( =\sqrt{4+49} \)
\( =\sqrt{53} \)
Similarly,
\( B C=\sqrt{(2+5)^{2}+(-3+5)^{2}} \)
\( =\sqrt{(7)^{2}+(2)^{2}} \)
\( =\sqrt{49+4} \)
\( =\sqrt{53} \)
\( C D=\sqrt{(4-2)^{2}+(4+3)^{2}} \)
\( =\sqrt{(2)^{2}+(7)^{2}} \)
\( =\sqrt{4+49} \)
\( =\sqrt{53} \)
\( D A=\sqrt{(-3-4)^{2}+(2-4)^{2}} \)
\( =\sqrt{(-7)^{2}+(-2)^{2}} \)
\( =\sqrt{49+4} \)
\( =\sqrt{53} \)
Diagonal \( \mathrm{AC}=\sqrt{(2+3)^{2}+(-3-2)^{2}} \)
\( =\sqrt{(5)^{2}+(-5)^{2}} \)
\( =\sqrt{25+25} \)
\( =\sqrt{50} \)
\( =\sqrt{25 \times 2} \)
\( =5 \sqrt{2} \)
Diagonal \( \mathrm{BD}=\sqrt{(4+5)^{2}+(4+5)^{2}} \)
\( =\sqrt{(9)^{2}+(9)^{2}} \)
\( =\sqrt{81+81} \)
\( =\sqrt{162} \)
\( =\sqrt{81 \times 2} \)
\( =9 \sqrt{2} \)
Here,
$A B=B C=C D=D A=\sqrt{53}$
Sides are equal but diagonals are not equal.
Therefore, $A (-3, 2), B (-5, -5), C (2, -3)$ and $D (4, 4)$ are the vertices of a rhombus.