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.

$\left(2,4\right)$, $\left(-3,4\right)$

Step-by-Step Solution

Answer

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

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

Step 1. Given.

Coordinates are $\left(2,4\right)$ and $\left(-3,4\right)$  

Step 2. Find the distance between the points 2 , 4 and − 3 , 4 .

The distance (d) between the points $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ 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 $\left(2,4\right)$ and $\left(-3,4\right)$ is:

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

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

Step 3. Find the midpoint of the segment with the endpoints at 2 , 4 and − 3 , 4 .

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

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

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

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