Question

Using the distance formula, prove that the points $A\left( -2,3 \right),B\left( 1,2 \right)\ and\ C\left( 7,0 \right)$ are collinear.

Answer 1

Hint: We will be using the concept of coordinate geometry to solve the problem. We will be using the fact that if three points A, B, C are collinear i.e. they lie in a line then,
AB + BC = AC

Complete step-by-step answer:

Now, we have been given three points as,
$A\left( -2,3 \right),B\left( 1,2 \right)\ and\ C\left( 7,0 \right)$

Now, if points are collinear then we have to prove that AB + BC = AC.
Now, we know that the distance between two points with coordinate $\left( {{x}_{1}},{{y}_{1}} \right)\ and\ \left( {{x}_{2}},{{y}_{2}} \right)$ is $AB=\sqrt{{{\left( {{x}_{1}}-{{x}_{2}} \right)}^{2}}+{{\left( {{y}_{1}}-{{y}_{2}} \right)}^{2}}}$.
Now, we have the distance AB as,
$\begin{align}
  & =\sqrt{{{\left( -2-1 \right)}^{2}}+{{\left( 3-2 \right)}^{2}}} \\
 & =\sqrt{{{\left( -3 \right)}^{2}}+{{\left( 1 \right)}^{2}}} \\
 & =\sqrt{9+1} \\
 & AB=\sqrt{10}units.........\left( 1 \right) \\
\end{align}$
Now, we have the distance BC as,
$\begin{align}
  & =\sqrt{{{\left( 1-7 \right)}^{2}}+{{\left( 2-0 \right)}^{2}}} \\
 & =\sqrt{{{\left( -6 \right)}^{2}}+{{\left( 2 \right)}^{2}}} \\
 & =\sqrt{36+4} \\
 & =\sqrt{40} \\
 & BC=2\sqrt{10}units.........\left( 2 \right) \\
\end{align}$
Now, we have the distance AC as,
$\begin{align}
  & =\sqrt{{{\left( -2-7 \right)}^{2}}+{{\left( 3-0 \right)}^{2}}} \\
 & =\sqrt{{{\left( -9 \right)}^{2}}+{{\left( 3 \right)}^{2}}} \\
 & =\sqrt{81+9} \\
 & =\sqrt{90} \\
 & AC=3\sqrt{10}units.........\left( 3 \right) \\
\end{align}$
Now, on adding (1) and (2) we have,
$\begin{align}
  & AB+BC=\sqrt{10}+2\sqrt{10}units \\
 & =3\sqrt{10}units \\
\end{align}$
Now, on equating this with equation (3) we have,
AB + BC = AC
So, we proved that the points A, B, C are collinear. Since, we have proved that,
AB + BC = AC

Note: To solve these type of question it is important to note that we have used a fact that if A, B, C are collinear then,
AB + BC = AC .
Also, it has to be noted that we have taken A, B, C as their order on graph.