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 the following sets of points are collinear.$(1, -1), (2, 1)$ and $(4, 5)$
Given:
Given vertices are $(1, -1), (2, 1)$ and $(4, 5)$.
To do:
We have to show that the given points are collinear.
Solution:
Let $A (1, -1), B (2, 1)$ and $C (4, 5)$ be the vertices of a triangle $ABC$.
We know that,
Three points are collinear if the area of the triangle formed by them is zero.
Area of a triangle with vertices $(x_1,y_1), (x_2,y_2), (x_3,y_3)$ is given by,
Area of $\Delta=\frac{1}{2}[x_{1}(y_{2}-y_{3})+x_{2}(y_{3}-y_{1})+x_{3}(y_{1}-y_{2})]$
Therefore,
Area of \( \Delta \mathrm{ABC}=\frac{1}{2}[1(1-5)+2(5+1)+4(-1-1)] \)
\( =\frac{1}{2} [1 \times(-4)+2 \times 6+4 \times(-2)] \)
\( =\frac{1}{2}[-4+12-8] \)
\( =\frac{1}{2} \times 0 \)
\( =0 \)
Here,
Area of \( \Delta \mathrm{ABC}=0 \)
Therefore, points \( \mathrm{A}, \mathrm{B} \) and \( \mathrm{C} \) are collinear.
Hence proved.
33 Views
