Chapter 1.1, Problem 56E

a .

To determine

To find: three points that divide the line segment joining the points $\left(x_1,y_1\right)=\left(1,-2\right)$ and $\left(x_2,y_2\right)=\left(4,-1\right)$ into four equal parts.

Answer to Problem 56E

The three points are $\left(\frac{5}{2},-\frac{3}{2}\right)$ , $\left(\frac{7}{4},-\frac{7}{4}\right)$ and $\left(\frac{13}{4},-\frac{5}{4}\right)$ .

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:

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

Calculation:

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 $\left(x_1,y_1\right)=\left(1,-2\right)$ and $\left(x_2,y_2\right)=\left(4,-1\right)$ .

  $\begin{array}{c}\text{Midpoint}=\left(\frac{1+4}{2},\frac{-2+\left(-1\right)}{2}\right)\\ =\left(\frac{5}{2},-\frac{3}{2}\right)\end{array}$

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 $\left(1,-2\right)$ and $\left(\frac{5}{2},-\frac{3}{2}\right)$ .

  $\begin{array}{c}\text{Midpoint}=\left(\frac{1+\frac{5}{2}}{2},\frac{-2+\left(-\frac{3}{2}\right)}{2}\right)\\ =\left(\frac{2+5}{4},\frac{-4-3}{4}\right)\\ =\left(\frac{7}{4},-\frac{7}{4}\right)\end{array}$

Now apply midpoint formula to the points $\left(\frac{5}{2},-\frac{3}{2}\right)$ and $\left(4,-1\right)$ .

  $\begin{array}{c}\text{Midpoint}=\left(\frac{\frac{5}{2}+4}{2},\frac{-\frac{3}{2}+\left(-1\right)}{2}\right)\\ =\left(\frac{5+8}{4},\frac{-3-2}{4}\right)\\ =\left(\frac{13}{4},-\frac{5}{4}\right)\end{array}$

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 $\left(\frac{5}{2},-\frac{3}{2}\right)$ , $\left(\frac{7}{4},-\frac{7}{4}\right)$ and $\left(\frac{13}{4},-\frac{5}{4}\right)$ .

b.

To determine

To find: three points that divide the line segment joining the points $\left(x_1,y_1\right)=\left(-2,-3\right)$ and $\left(x_2,y_2\right)=\left(0,0\right)$ into four equal parts.

Answer to Problem 56E

The three points are $\left(-1,-\frac{3}{2}\right)$ , $\left(-\frac{3}{2},-\frac{9}{4}\right)$ and $\left(-\frac{1}{2},-\frac{3}{4}\right)$ .

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:

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

Calculation:

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 $\left(x_1,y_1\right)=\left(-2,-3\right)$ and $\left(x_2,y_2\right)=\left(0,0\right)$ .

  $\begin{array}{c}\text{Midpoint}=\left(\frac{-2+0}{2},\frac{-3+0}{2}\right)\\ =\left(-\frac{2}{2},-\frac{3}{2}\right)\\ =\left(-1,-\frac{3}{2}\right)\end{array}$

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 $\left(-2,-3\right)$ and $\left(-1,-\frac{3}{2}\right)$ .

  $\begin{array}{c}\text{Midpoint}=\left(\frac{-2+\left(-1\right)}{2},\frac{-3+\left(-\frac{3}{2}\right)}{2}\right)\\ =\left(\frac{-3}{2},\frac{-6-3}{4}\right)\\ =\left(-\frac{3}{2},-\frac{9}{4}\right)\end{array}$

Now apply midpoint formula to the points $\left(-1,-\frac{3}{2}\right)$ and $\left(0,0\right)$ .

  $\begin{array}{c}\text{Midpoint}=\left(\frac{-1+0}{2},\frac{-\frac{3}{2}+0}{2}\right)\\ =\left(-\frac{1}{2},-\frac{3}{4}\right)\end{array}$

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 $\left(-1,-\frac{3}{2}\right)$ , $\left(-\frac{3}{2},-\frac{9}{4}\right)$ and $\left(-\frac{1}{2},-\frac{3}{4}\right)$ .