Question

In the following exercises, find the distance between the points. Write the answer in exact form and then find the decimal approximation, rounded to the nearest tenth if needed.

$(-1,-2)$and$(-3,4)$

Short answer

The exact distance is $\sqrt{40}$and the approximated distance is $6.3$.

Step-by-step solution

Step 1. Given Information 

We are given points$(-1,-2),(-3,4)$.

The distance formula states that the distance between points $\left(x_1,y_1\right)$and $\left(x_2,y_2\right)$is,

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

Step 2. Applying the formula 

Consider the point $(-1,-2)$as $\left(x_1,y_1\right)$and $(-3,4)$as $\left(x_2,y_2\right)$.

Substituting the values in the distance formula, we get

$$\begin{gathered} d=\sqrt{[-3-(-1){]}^2+[4-(-2){]}^2} \\ d=\sqrt{(-3+1{)}^2+(4+2{)}^2} \\ d=\sqrt{(-2{)}^2+(6{)}^2} \\ d=\sqrt{4+36} \\ d=\sqrt{40} \\ d\approx 6.3 \end{gathered}$$

The exact distance is $\sqrt{40}$ and the approximated distance is $6.3$.