Chapter 1.1, Problem 55E

To determine

To find: three points that divide the line segment joining the points $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ into four equal parts.

Answer to Problem 55E

The three points are $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$ , $\left(\frac{3x_1+x_2}{4},\frac{3y_1+y_2}{4}\right)$ and $\left(\frac{x_1+3x_2}{4},\frac{y_1+3y_2}{4}\right)$ .

Explanation of Solution

Concept Used:

Midpoint Formula: Midpoint of the line segment joining the two points $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ is given by:

  $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$

Midpoint formula is used to divide a line segment joining two points in two equal parts.

So applying Midpoint formula on the line segment joining the two points $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$

Now apply midpoint formula to each of the two equal parts obtained above to obtain 4 equal parts.

First apply midpoint formula to the points $\left(x_1,y_1\right)$ and $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$ .

  $\begin{array}{c}\text{Midpoint}=\left(\frac{x_1+\frac{x_1+x_2}{2}}{2},\frac{y_1+\frac{y_1+y_2}{2}}{2}\right)\\ =\left(\frac{2x_1+x_1+x_2}{4},\frac{2y_1+y_1+y_2}{4}\right)\\ =\left(\frac{3x_1+x_2}{4},\frac{3y_1+y_2}{4}\right)\end{array}$

Now apply midpoint formula to the points $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$ and $\left(x_2,y_2\right)$ .

  $\begin{array}{c}\text{Midpoint}=\left(\frac{\frac{x_1+x_2}{2}+x_2}{2},\frac{\frac{y_1+y_2}{2}+y_2}{2}\right)\\ =\left(\frac{x_1+x_2+2x_2}{4},\frac{y_1+y_2+2y_2}{4}\right)\\ =\left(\frac{x_1+3x_2}{4},\frac{y_1+3y_2}{4}\right)\end{array}$

Conclusion:

Three points that divide the line segment joining the points $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ into four equal parts are $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$ , $\left(\frac{3x_1+x_2}{4},\frac{3y_1+y_2}{4}\right)$ and $\left(\frac{x_1+3x_2}{4},\frac{y_1+3y_2}{4}\right)$ .