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.
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