Question

Find the distance between the points (
–
4,
–
8) and (9,3).
Write your answer as a whole number or a fully simplified radical expression. Do not round.

Answer 1

To find the distance between the points \((-4, -8)\) and \((9, 3)\), we can use the distance formula, which is given by:

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

Here, \((x_1, y_1) = (-4, -8)\) and \((x_2, y_2) = (9, 3)\).

Substituting the coordinates into the distance formula:

\[
d = \sqrt{(9 - (-4))^2 + (3 - (-8))^2}
\]

This simplifies to:

\[
d = \sqrt{(9 + 4)^2 + (3 + 8)^2}
\]

Which further simplifies to:

\[
d = \sqrt{(13)^2 + (11)^2}
\]

Calculating the squares:

\[
d = \sqrt{169 + 121}
\]

Adding those values together:

\[
d = \sqrt{290}
\]

To express \( \sqrt{290} \) in its simplest form, we check if 290 can be factored into perfect squares:

The prime factorization of 290 is \(2 \times 145\) and since \(145\) factors into \(5 \times 29\), the prime factorization of \(290\) is \(2 \times 5 \times 29\), and there are no perfect square factors.

Thus, \( \sqrt{290} \) is already in its simplest form:

The distance between the points \((-4, -8)\) and \((9, 3)\) is

\[
\boxed{\sqrt{290}}
\]