Chapter 1.1, Problem 50E

a.

To determine

Find the points that divide the line segment.

Answer to Problem 50E

  $\boxed{\frac{5}{2},\frac{1}{2}}$ , $\boxed{\frac{7}{4},\frac{5}{4}}$ , $\boxed{\frac{13}{4},\frac{-1}{4}}$

Explanation of Solution

Given information:

To find the points that divides the line segment joining the given points into four equal parts.

Using the midpoint formula.

  $\left(1,-2\right),\left(4,-1\right)$

Calculation:

The one end of line segment $\left(x_1,y_1\right)$ and other end $\left(x_2,y_2\right)$ having midpoint $\left(x_m,y_m\right)$

The coordinate value of $\left(x_1,y_1\right)$ and midpoint $\left(x_m,y_m\right)$ are given.

Now find coordinate value of $\left(x_2,y_2\right)$ by using midpoint formula,

  $\begin{array}{l}{x}_m=\frac{\left(x_1+x_2\right)}{2}\\ 2x_m=\left(x_1+x_2\right)\\ \boxed{x_2=2x_m-x_1}\end{array}$

  $\begin{array}{l}po\mathrm{int}1=\frac{\left(x_1+x_2\right)}{2},\frac{\left(y_1+y_2\right)}{2}\\ po\mathrm{int}2=\frac{\left(2x_1+x_1+x_2\right)}{4},\frac{\left(2y_1+y_1+y_2\right)}{4}\\ po\mathrm{int}3=\frac{\left(x_1+x_2+2x_2\right)}{4},\frac{\left(y_1+y_2+2y_2\right)}{4}\end{array}$

We have given the coordinates of two end of line segment.

Substitute the value

  $\left(1,-2\right),\left(4,-1\right)$ in above expression.

  $\begin{array}{l}po\mathrm{int}1=\frac{1+4}{2},\frac{2-1}{2}\\ =\boxed{\frac{5}{2},\frac{1}{2}}\\ po\mathrm{int}2=\frac{1+\frac{5}{2}}{2},\frac{2+\frac{1}{2}}{2}\\ =\frac{2+5}{4},\frac{4+1}{4}\\ =\boxed{\frac{7}{4},\frac{5}{4}}\\ po\mathrm{int}3=\frac{4+\frac{5}{2}}{2},\frac{-1+\frac{1}{2}}{2}\\ =\frac{8+5}{4},\frac{-2+1}{4}\\ =\boxed{\frac{13}{4},\frac{-1}{4}}\end{array}$

Hence the coordinates of other end of line segment $\boxed{\frac{5}{2},\frac{1}{2}}$ , $\boxed{\frac{7}{4},\frac{5}{4}}$ , $\boxed{\frac{13}{4},\frac{-1}{4}}$

b.

To determine

Find the points that divide the line segment.

Answer to Problem 50E

  $\boxed{-1,\frac{-3}{2}}$ , $\boxed{\frac{-3}{2},\frac{-9}{4}}$ , $\boxed{\frac{-1}{2},\frac{-3}{4}}$

Explanation of Solution

Given information:

To find the points that divides the line segment joining the given points into four equal parts.

Using the midpoint formula.

  $\left(-2,-3\right)and\left(0,0\right)$

Calculation:

The one end of line segment $\left(x_1,y_1\right)$ and other end $\left(x_2,y_2\right)$ having midpoint $\left(x_m,y_m\right)$

The coordinate value of $\left(x_1,y_1\right)$ and midpoint $\left(x_m,y_m\right)$ are given.

Now find coordinate value of $\left(x_2,y_2\right)$ by using midpoint formula,

  $\begin{array}{l}{x}_m=\frac{\left(x_1+x_2\right)}{2}\\ 2x_m=\left(x_1+x_2\right)\\ \boxed{x_2=2x_m-x_1}\end{array}$

  $\begin{array}{l}po\mathrm{int}1=\frac{\left(x_1+x_2\right)}{2},\frac{\left(y_1+y_2\right)}{2}\\ po\mathrm{int}2=\frac{\left(2x_1+x_1+x_2\right)}{4},\frac{\left(2y_1+y_1+y_2\right)}{4}\\ po\mathrm{int}3=\frac{\left(x_1+x_2+2x_2\right)}{4},\frac{\left(y_1+y_2+2y_2\right)}{4}\end{array}$

We have given the coordinates of two end of line segment.

Substitute the value

  $\left(-2,-3\right)and\left(0,0\right)$ in above expression.

  $\begin{array}{l}po\mathrm{int}1=\frac{-2+0}{2},\frac{-3+0}{2}\\ =\boxed{-1,\frac{-3}{2}}\\ po\mathrm{int}2=\frac{-2+\left(-1\right)}{2},\frac{-3+\left(\frac{-3}{2}\right)}{2}\\ =\frac{-2-1}{2},\frac{-6-3}{4}\\ =\boxed{\frac{-3}{2},\frac{-9}{4}}\\ po\mathrm{int}3=\frac{0+\left(-1\right)}{2},\frac{0+\left(\frac{-3}{2}\right)}{2}\\ =\boxed{\frac{-1}{2},\frac{-3}{4}}\end{array}$

Hence the coordinates of other end of line segment, $\boxed{-1,\frac{-3}{2}}$ , $\boxed{\frac{-3}{2},\frac{-9}{4}}$ , $\boxed{\frac{-1}{2},\frac{-3}{4}}$