Q54. - 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.

$\left(-1,-3\right), \left(3,5\right)$

Step-by-Step Solution

Answer

The distance between the points $\left(-1,-3\right)$ and $\left(3,5\right)$ is 8.94.

The midpoint of the line segment with the endpoints at $\left(-1,-3\right)$ and $\left(3,5\right)$ is $\left(1,1\right)$.

Step 1. Given.

Coordinates are $\left(-1,-3\right)$ and $\left(3,5\right)$

Step 2. Find the distance between the points − 1 , − 3 and 3 , 5 .

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 $\left(-1,-3\right)$ and $\left(3,5\right)$ is:

width="328" height="268" style="max-width: none; vertical-align: -135px;" $$\begin{gathered} d=\sqrt{{\left(3-\left(-1\right)\right)}^2+{\left(5-\left(-3\right)\right)}^2} \\ =\sqrt{{\left(3+1\right)}^2+{\left(5+3\right)}^2} \\ =\sqrt{4^2+8^2} \\ =\sqrt{16+64} \\ =\sqrt{80} \\ =\sqrt{16\times 5} \\ =4\sqrt{5} \\ =4\left(2.236068\right) \\ =8.944272 \\ \approx 8.94\left(\text{rounded to the nearest hundredth}\right) \end{gathered}$$

Therefore, the distance (d) between the points $\left(-1,-3\right)$ and $\left(3,5\right)$ is $8.94$.

Step 3. Find the midpoint of the segment with the endpoints at − 1 , − 3 and 3 , 5 .

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 $\left(-1,-3\right)$ and $\left(3,5\right)$ is given by:

$$\begin{gathered} M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right) \\ =\left(\frac{-1+3}{2},\frac{-3+5}{2}\right) \\ =\left(\frac{2}{2},\frac{2}{2}\right) \\ =\left(1,1\right) \end{gathered}$$

Therefore, the midpoint of the line segment with the endpoints at $\left(-1,-3\right)$ and $\left(3,5\right)$ is $\left(1,1\right)$.