Chapter 5.1, Problem 54PPE

To determine

Todetermine:whether the points of given set lie on the same line.

Answer to Problem 54PPE

The points $G\left(1,-2\right)$ , $H\left(-1,-5\right)$ , $I\left(5,4\right)$ lie on same line.

Explanation of Solution

Given information:

The given set of points is $G\left(1,-2\right)$ , $H\left(-1,-5\right)$ , $I\left(5,4\right)$ .

Conceptused:

The slopes of two segments of a line is same. The slope of line passing through points $\left(x_1,y_1\right)$ , $\left(x_2,y_2\right)$ is given by $m=\frac{y_2-y_1}{x_2-x_1}$ .

Suppose point $A\left(x_1,y_1\right)$ , $B\left(x_2,y_2\right)$ , $C\left(x_3,y_3\right)$ do not lie on a line as shown in Figure (a) here. Then the slope of lineAB is not equal to slope of line AC , i.e. $m_{AB}\ne m_{AC}$ or $\frac{y_2-y_1}{x_2-x_1}\ne \frac{y_3-y_1}{x_3-x_1}$ .

  

If points $A\left(x_1,y_1\right)$ , $B\left(x_2,y_2\right)$ , $C\left(x_3,y_3\right)$ lie on a line as shown in Figure (b) then the slope of line AB is not equal to slope of line AC , i.e. $m_{AB}=m_{AC}$ or $\frac{y_2-y_1}{x_2-x_1}=\frac{y_3-y_1}{x_3-x_1}$ .

For given points $m_{AB}=\frac{y_2-y_1}{x_2-x_1}=\frac{2-3}{4-1}=-\frac{1}{3}$ , and $m_{AC}=\frac{y_3-y_1}{x_3-x_1}=\frac{4-3}{-2-1}=-\frac{1}{3}$ .

Now, the given set of points is $G\left(1,-2\right)$ , $H\left(-1,-5\right)$ , $I\left(5,4\right)$ .

The slope of line segment GHis $m_{GH}=\frac{-5-(-2)}{-1-1}=\frac{-5+2}{-2}=\frac{-3}{-2}=\frac{3}{2}$ .

The slope of line segment GIis $m_{GI}=\frac{4-(-2)}{5-1}=\frac{4+2}{4}=\frac{6}{4}=\frac{3}{2}$ .

Since, $m_{GH}=m_{GI}$ , therefore thepoints $G\left(1,-2\right)$ , $H\left(-1,-5\right)$ , $I\left(5,4\right)$ lie on same line.

Conclusion:

Thepoints $G\left(1,-2\right)$ , $H\left(-1,-5\right)$ , $I\left(5,4\right)$ lie on same line.