Question

Use the rectangular coordinate system to find the distance between the points $(5,3)$ and $(-3,-3)$.

Short answer

The distance between the given points is $10$.

Step-by-step solution

Step 1. Plot the points 

Plot the two points. Connect the two points with a line and then draw a right triangle.

Step 2. Find the lengths 

Find the length of each leg.

The rise is six units and the run is eight units. So the right triangle can be shown as

Step 3. Use Pythagorean Theorem 

Use the Pythagorean Theorem, $a^2+b^2=c^2$, to find $d$, the distance between the two points.

On substituting $6$for $a$, $8$for $b$, and $d$for $c$we get

$\begin{array}{rcl}{6}^2+8^2& =& d^2\\ 36+64& =& d^2\\ 100& =& d^2\end{array}$

Step 4. Solve for d 

As $d^2=100$, so using the square root property we get

$$\begin{gathered} d=\pm \sqrt{100} \\ d=\pm 10 \end{gathered}$$

But the value of distance cannot be negative, so we eliminate $d=-10$.

So the distance between the points is $d=10$.