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.

$\mathbf{(}\mathbf{-}\mathbf{1}\mathbf{,}\mathbf{-}\mathbf{3}\mathbf{)}$, $\mathbf{(}\mathbf{3}\mathbf{,}\mathbf{5}\mathbf{)}$

Step-by-Step Solution

Answer

The distance between the points $(-1,-3)$ and $(3,5)$ is $\mathbf{8}\mathbf{.94}$.

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

Step2. Given

Coordinates are $(-1,-3)$ and $(3,5)$

Step2. Find the distance between the points ( − 1 , − 3 ) and ( 3 , 5 ) .

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

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

Therefore, the distance ($d$) between the points $(-1,-3)$ and $(3,5)$ is:

$\begin{array}{l}d=\sqrt{{(3-(-1))}^2+{(5-(-3))}^2}\\ =\sqrt{{(3+1)}^2+{(5+3)}^2}\\ =\sqrt{4^2+8^2}\\ =\sqrt{16+64}\\ =\sqrt{80}\\ =\sqrt{16\times 5}\\ =4\sqrt{5}\\ =4\left(2.236068\right)\\ =8.944272\\ \approx 8.94\left(\text{rounded to the nearest hundredth}\right)\end{array}$

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

Step3. Find the midpoint of the segment with the endpoints at ( − 1 , − 3 ) and ( 3 , 5 ) .

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 $(-1,-3)$ and $(3,5)$ is given by:

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

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