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.

$\left(1,5\right),\left(-4,-5\right)$

Short answer

The distance between the points$\left(1,5\right)$ and$\left(-4,-5\right)$ is $11.18$.

The midpoint of the line segment with the endpoints at$\left(1,5\right)$ and$\left(-4,-5\right)$ is $\left(-1.5,0\right)$.

Step-by-step solution

Step 1. Given.

Coordinates are$\left(1,5\right)$ and$\left(-4,-5\right)$

Step 2. Find the distance between the points 1,5 and −4,−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$\left(1,5\right)$ and$\left(-4,-5\right)$ is:

$$\begin{gathered} d=\sqrt{{\left(-4-1\right)}^2+{\left(-5-5\right)}^2} \\ =\sqrt{{\left(-5\right)}^2+{\left(-10\right)}^2} \\ =\sqrt{5^2+{10}^2} \\ =\sqrt{25+100} \\ =\sqrt{125} \\ =\sqrt{25\times 5} \\ =5\sqrt{5} \\ =5\left(2.2361\right) \\ =11.1805 \\ \approx 11.18\left(\text{roundedtothenearesthundredth}\right) \end{gathered}$$

Therefore, the distance (d) between the points$\left(1,5\right)$ and$\left(-4,-5\right)$ is 11.18.

Step 3. Find the midpoint of the segment with the endpoints at 1,5 and −4,−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$\left(1,5\right)$ and$\left(-4,-5\right)$ is given by:

$$\begin{gathered} M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right) \\ =\left(\frac{1+\left(-4\right)}{2},\frac{5+\left(-5\right)}{2}\right) \\ =\left(\frac{1-4}{2},\frac{5-5}{2}\right) \\ =\left(\frac{-3}{2},\frac{0}{2}\right) \\ =\left(-1.5,0\right) \end{gathered}$$

Therefore, the midpoint of the line segment with the endpoints at$\left(1,5\right)$ and$\left(-4,-5\right)$ is $\left(-1.5,0\right)$.