Question

Find the distance between the points (
–
4,
–
3) and (0,3).
Write your answer as a whole number or a fully simplified radical expression. Do not round.

Answer 1

To find the distance between two points \((x_1, y_1)\) and \((x_2, y_2)\), we use the distance formula:

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

Let's identify the coordinates:
- Point 1: \((-4, -3)\) which gives \(x_1 = -4\), \(y_1 = -3\)
- Point 2: \((0, 3)\) which gives \(x_2 = 0\), \(y_2 = 3\)

Now, substitute the values into the distance formula:

\[
d = \sqrt{(0 - (-4))^2 + (3 - (-3))^2}
\]

Calculating each part step by step:

1. Calculate \(x_2 - x_1\):
\[
0 - (-4) = 0 + 4 = 4
\]

2. Calculate \(y_2 - y_1\):
\[
3 - (-3) = 3 + 3 = 6
\]

Now plug these values back into the formula:

\[
d = \sqrt{(4)^2 + (6)^2}
\]

Calculate the squares:

\[
d = \sqrt{16 + 36}
\]
\[
d = \sqrt{52}
\]

Now simplify \(\sqrt{52}\). We can factor out \(4\):

\[
\sqrt{52} = \sqrt{4 \times 13} = \sqrt{4} \cdot \sqrt{13} = 2\sqrt{13}
\]

Thus, the distance between the points \((-4, -3)\) and \((0, 3)\) is

\[
\boxed{2\sqrt{13}}
\]