Question

find the distance between the points (0, -8) and (-8, 3). write your answer as a whole number or a fully simplified radical expression. do not round. units

Explanation:

Step1: Recall the distance formula

The distance \( d \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \).
Here, \( x_1 = 0 \), \( y_1 = -8 \), \( x_2 = -8 \), \( y_2 = 3 \).

Step2: Substitute the values into the formula

First, calculate the differences in \( x \)-coordinates and \( y \)-coordinates:
\( x_2 - x_1 = -8 - 0 = -8 \)
\( y_2 - y_1 = 3 - (-8) = 3 + 8 = 11 \)

Then, substitute these into the distance formula:
\( d = \sqrt{(-8)^2 + 11^2} \)

Step3: Simplify the expression inside the square root

Calculate the squares:
\( (-8)^2 = 64 \)
\( 11^2 = 121 \)

Add the results:
\( 64 + 121 = 185 \)

So, \( d = \sqrt{185} \) (since 185 cannot be simplified further as a radical because its prime factorization is \( 5 \times 37 \), and neither 5 nor 37 are perfect squares).

Answer:

\( \sqrt{185} \)