Question

Using the method of slope, show that the points $A\left( 4,8 \right),B\left( 5,12 \right),C\left( 9,28 \right)$are collinear.

Answer 1

Hint: We will be using the concept of coordinate geometry. We will find the slope of any two pairs of points and we know that if points are collinear, they line on the same line and hence, their slope will be the same.

Complete step-by-step answer:

Now, we have three points as,
$A\left( 4,8 \right),B\left( 5,12 \right),C\left( 9,28 \right)$

Now, we know that the slope between any two points $\left( {{x}_{1}},{{y}_{1}} \right)\ and\ \left( {{x}_{2}},{{y}_{2}} \right)$is,
$slope=\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}$
Now, we find the slope of AB. So, we have,
$\begin{aligned} & slope\ of\ AB=\dfrac{12-8}{5-4} \\\ & =4 \\\ \end{aligned}$
Now, we will find the slope of BC. So, we have,
$\begin{aligned} & slope\ of\ BC=\dfrac{28-12}{9-5} \\\ & =\dfrac{16}{4} \\\ & =4 \\\ \end{aligned}$
Now, we can see that the slope of AB is equal to that of BC. So, we can say that the points A, B and C lie on the same line. So, these points are collinear.

Note: To solve these types of questions it is important to note that we have used the fact that if three points are collinear then the slope of any two pairs will be the same.