Question

The points A (1, 2), B (2, 4) and C (4, 8) form an:
1. Isosceles triangle
2. Equilateral triangle
3. Straight line
4. Right-angled triangle

Answer 1

Hint: For solving this question you should know about the distance between two points and the slope between two points. Here for solving this problem, we will find the distance between all the points and then we will find the slope. And then we will find the correct answer for this statement.

Complete step-by-step solution:
It is asked to us to find the right relation of a figure if the points A (1, 2), B (2, 4) and C (4, 8) form that. As we know that the distance between two points is calculated by the formula,
Distance = $\sqrt{{{\left( {{x}_{2}}-{{x}_{1}} \right)}^{2}}+{{\left( {{y}_{2}}-{{y}_{1}} \right)}^{2}}}$
And the slope of that point is calculated as,
Slope = $\dfrac{{{x}_{2}}-{{x}_{1}}}{{{y}_{2}}-{{y}_{1}}}$
So, the distance between all the points are as follows:
Distance between A and B is given by,
$\begin{align}
  & \sqrt{{{\left( 2-1 \right)}^{2}}+{{\left( 4-2 \right)}^{2}}} \\
 & =\sqrt{1+4}=\sqrt{5} \\
\end{align}$
Distance between B and C is given by,
$\begin{align}
  & \sqrt{{{\left( 4-2 \right)}^{2}}+{{\left( 8-4 \right)}^{2}}} \\
 & =\sqrt{4+16}=\sqrt{20} \\
\end{align}$
Distance between A and C is given by,
$\begin{align}
  & \sqrt{{{\left( 1-4 \right)}^{2}}+{{\left( 2-8 \right)}^{2}}} \\
 & =\sqrt{9+36}=\sqrt{45} \\
\end{align}$
Now let us find the slopes:
So, slope of AB = $\dfrac{4-2}{2-1}=2$
Slope of BC = $\dfrac{8-4}{4-2}=2$
Ans the slope of CA = $\dfrac{2-8}{1-4}=2$
Here all the slopes are the same and the distance is $\sqrt{20}+\sqrt{5}=\sqrt{45}$ So, the three points form a straight line or the three points are collinear.
Hence the correct option is answer 3.

Note: While solving these types of questions you have to be careful about the selection of the points and finding the distance between them carefully. The points can interchange with each other, but they cannot change for any point. So, always take fixed points.