Question

A graph shows quadrant 1 with the x-axis and y-axis ranging from 0 to positive 9.
Apply the Pythagorean Theorem to find the length between the two points. Round your answer to the nearest hundredth. (3, 2) 7, 8)

Answer 1

To find the distance between the two points \((3, 2)\) and \((7, 8)\), we can apply the distance formula derived from the Pythagorean theorem. The distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

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

For the points \((3, 2)\) and \((7, 8)\):
- \(x_1 = 3\)
- \(y_1 = 2\)
- \(x_2 = 7\)
- \(y_2 = 8\)

Now, substitute these values into the distance formula:

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

Calculate the differences:

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

Now calculate the squares:

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

Combine the results:

\[
d = \sqrt{52}
\]

Now, simplify \(\sqrt{52}\):

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

Now, calculate \(2\sqrt{13}\). The approximate value of \(\sqrt{13} \approx 3.60555\). Thus,

\[
2\sqrt{13} \approx 2 \times 3.60555 = 7.2111
\]

Rounding this to the nearest hundredth, the length between the two points is approximately:

\[
\boxed{7.21}
\]