Question

Use the image to answer the question.

A graph shows quadrant 1 with the x-axis and y-axis ranging from 0 to positive 9, 3,2 and 7,8 are highlighted.

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

Answer 1

To find the distance between the points (3, 2) and (7, 8) using the Pythagorean Theorem, we can use the distance formula, which is derived from the Pythagorean Theorem:

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

Where:
- \((x_1, y_1)\) = (3, 2)
- \((x_2, y_2)\) = (7, 8)

Now, we can substitute the coordinates into the formula:

\[
d = \sqrt{(7 - 3)^2 + (8 - 2)^2}
\]

Calculating the differences:

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

Now squaring the differences:

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

\[
d = \sqrt{52}
\]

Now we can simplify \(\sqrt{52}\):

\[
d = \sqrt{4 \cdot 13} = 2\sqrt{13}
\]

Using a calculator, we approximate \(\sqrt{13}\):

\[
\sqrt{13} \approx 3.60555
\]

Therefore:

\[
d \approx 2 \times 3.60555 \approx 7.2111
\]

Rounding to the nearest hundredth:

\[
d \approx 7.21
\]

Thus, the length between the two points is approximately **7.21**.