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:
\[ \boxed{7.21} \]