Q53. - Algebra 1

Question

Find the distance between points with the given coordinates and the midpoint of the segment with the given endpoints. Round to the nearest hundredth if necessary.

$\mathbf{(}\mathbf{2}\mathbf{,}\mathbf{4}\mathbf{)}$, $\mathbf{(}\mathbf{-}\mathbf{3}\mathbf{,}\mathbf{4}\mathbf{)}$

Step-by-Step Solution

Answer

The distance between the points $(2,4)$ and $(-3,4)$ is 5.

The midpoint of the line segment with the endpoints at $(2,4)$ and $(-3,4)$ is $\mathbf{(}\mathbf{-}\mathbf{0}\mathbf{.5}\mathbf{,}\mathbf{4}\mathbf{)}$

Step1. Given

Coordinates are $(2,4)$ and $(-3,4)$

Step2. Find the distance between the points ( 2 , 4 ) and ( − 3 , 4 ) .

The distance ($d$) between the points $(x_1,y_1)$ and $(x_2,y_2)$ is given by:

$d=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$

Therefore, the distance ($d$) between the points $(2,4)$ and $(-3,4)$ is:

$\begin{array}{l}d=\sqrt{{\left(-3-2\right)}^2+{\left(4-4\right)}^2}\\ =\sqrt{{\left(-5\right)}^2+{\left(0\right)}^2}\\ =\sqrt{25+0}\\ =\sqrt{25}\\ =\sqrt{5^2}\\ =5\end{array}$

Therefore, the distance ($d$) between the points $(2,4)$ and $(-3,4)$ is 5.

Step3. Find the midpoint of the segment with the endpoints at ( 2 , 4 ) and ( − 3 , 4 ) .

The midpoint formula states that the midpoint of a line segment with endpoints at $(x_1,y_1)$ and $(x_2,y_2)$ is given by $M=\left(\frac{{\displaystyle x_1+x_2}}{{\displaystyle 2}},\frac{{\displaystyle y_1+y_2}}{{\displaystyle 2}}\right)$.

The midpoint ($M$) of the line segment with endpoints at $(2,4)$ and $(-3,4)$ is given by:

$\begin{array}{l}M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)\\ =\left(\frac{2+(-3)}{2},\frac{4+4}{2}\right)\\ =\left(\frac{2-3}{2},\frac{8}{2}\right)\\ =\left(\frac{-1}{2},4\right)\\ =(-0.5,4)\end{array}$

Therefore, the midpoint of the line segment with the endpoints at $(2,4)$ and $(-3,4)$ is $(-0.5,4)$.