Question

Apply the Pythagorean Theorem to find the length between the two points. Round your answer to the nearest hundredth, if necessary.

A: (2, -3)

B: (5, 5)

Answer 1

To find the length between the two points \( A(2, -3) \) and \( B(5, 5) \), we can use the distance formula derived from the Pythagorean Theorem. The distance \( d \) between the two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:

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

For the points \( A(2, -3) \) and \( B(5, 5) \):

- \( x_1 = 2 \)
- \( y_1 = -3 \)
- \( x_2 = 5 \)
- \( y_2 = 5 \)

Now, substitute these values into the distance formula:

\[
d = \sqrt{(5 - 2)^2 + (5 - (-3))^2}
\]

Calculating the differences:

- \( 5 - 2 = 3 \)
- \( 5 - (-3) = 5 + 3 = 8 \)

Now substitute these values back into the formula:

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

Calculating the squares:

- \( (3)^2 = 9 \)
- \( (8)^2 = 64 \)

Now add the squares:

\[
d = \sqrt{9 + 64} = \sqrt{73}
\]

Now we calculate the square root of 73:

\[
d \approx 8.544
\]

Rounding to the nearest hundredth:

\[
d \approx 8.54
\]

Thus, the length between the two points \( A(2, -3) \) and \( B(5, 5) \) is approximately \( \boxed{8.54} \).