Midpoint and Distance Formulas

Help Questions

Math › Midpoint and Distance Formulas

Questions 1 - 10

Find the midpoint of the line segment that has endpoints at $(7, 0) \text{ and }(8,9)$ .

$\left(\frac{15}{2}, \frac{9}{2}\right)$

$\left(\frac{1}{2}, -\frac{9}{2}\right)$

$\left(\frac{7}{2}, \frac{17}{2}\right)$

Explanation

To find the midpoint of a line segment, take the average of the x-coordinates and the average of the y-coordinates.

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

$\text{Midpoint}=\left(\frac{7+8}{2}, \frac{0+9}{2}\right)=\left(\frac{15}{2}, \frac{9}{2}\right)$

What is the midpoint of a line segment with the end points and ?

Explanation

In order to solve for the midpoint of a line segment when the end points are given, the x and y values must be averaged.

A formula for this would be:
$midpoint=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$

This probem may be quickly solved for by substituting in the given information.

$midpoint=(\frac{1+7}{2},\frac{2+8}{2})$

$midpoint=(\frac{8}{2},\frac{10}{2})$

$midpoint={\color{Blue} (4,5)}$

Find the length of the line connecting the following two points. Simplify your answer.

and

$5\sqrt{5}$

$4\sqrt{5}$

$\sqrt{123}$

$\sqrt{127}$

Explanation

To solve this problem we need to remember the distance formula for points on a coordinate plane:

$d=\sqrt{(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}}$

In this case, and

$d=\sqrt{(12-2)^{2}+(6-1)^{2}}$

$d=\sqrt{10^{2}+5^{2}}$

$d=\sqrt{100+25}=\sqrt{125}=5\sqrt{5}$

Find the length of the line with the following endpoints:

$\sqrt{29}$

$2\sqrt{27}$

$\sqrt{26}$

Explanation

In order to find the distance between two points on a line, use the formula below:

$\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}$

Plug in the points (2,1) and (7,3) and solve:

$\sqrt{(7-2)^{2}+(3-1)^{2}}=\sqrt{(5)^{2}+(2)^{2}}=\sqrt{25+4}=\sqrt{29}$

This gives a final answer of $\sqrt{29}$

$\textup{Find the distance between the midpoint of the line segment}\ \textup{ between points } (-1,-1)\textup{ and } (2,3) \textup{ and the midpoint of the line}\\textup{ segment between points }(-1,-1)\textup{ and (2,-1).}$

$\textup{None of the above.}$

$\frac{1}{3}$

$\frac{2}{3}$

$\textup{1.5}$

$\textup{4}$

Explanation

$\textup{The midpoint of the first line segment has coordinates }\\left(\frac{-1+2}{2},\frac{-1+3}{2} \right )=\left(0.5,1 \right) \textup{and the midpoint of the second}\\textup{ line segment has coordinates }\left(\frac{-1+2}{2},\frac{-1-1}{2} \right )=\left(0.5,-1 \right ).$

$\textup{The segment that joins the two midpoints is on a vertical line, }\\textup{therefore its length is the absolute difference between the two}\\textup{midpoints' }y\textup{-coordinates }\left | 1-(-1)\right |=2.$

$\textup{Alternatively, one can draw all the points and segments and find}\\textup{the distance by exploiting the similarity of the two right triangles.}$

Find the midpoint of a line segment with the following endpoints:

Explanation

Finding the midpoint of a line segment only requires that we find the midpoint, or average of both the x and y components. In order to do that, we use the following formula:

$(\frac{x_{1}+x_{2}}{2},\frac{y_{1}+y_{2}}{2})$

For the points (2,6) and (12,12) plug in the numbers and solve:

$(\frac{2+12}{2},\frac{6+12}{2})=(\frac{14}{2},\frac{18}{2})=(7,9)$

This gives a final answer of

Find the midpoint of a line segment with and as its endpoints.

Explanation

In order to solve for the midpoint of a line segment when the end points are given, the x and y values must be averaged.

A formula for this would be:
$midpoint=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$

This probem may be quickly solved for by substituting in the given information.

$midpoint=(\frac{0+10}{2},\frac{0+4}{2})$

$midpoint=(\frac{10}{2},\frac{4}{2})$

$midpoint={\color{Blue} (5,2)}$

Find the midpoint of the line segment with the following endpoints:

Explanation

Finding the midpoint of a line segment only requires that we find the midpoint, or average of both the x and y components. In order to do that, we use the following formula:

$(\frac{x_{1}+x_{2}}{2},\frac{y_{1}+y_{2}}{2})$