To find the distance between the points \((7, 8)\) and \((3, 2)\) using the Pythagorean theorem, we can use the distance formula, which is derived from the theorem. The distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
In this case, we have:
- \((x_1, y_1) = (7, 8)\)
- \((x_2, y_2) = (3, 2)\)
Now, substituting the values into the formula:
-
Calculate the differences:
- \(x_2 - x_1 = 3 - 7 = -4\)
- \(y_2 - y_1 = 2 - 8 = -6\)
-
Square these differences:
- \((-4)^2 = 16\)
- \((-6)^2 = 36\)
-
Add the squares:
- \(16 + 36 = 52\)
-
Take the square root:
- \(d = \sqrt{52}\)
To simplify \(\sqrt{52}\): \[ \sqrt{52} = \sqrt{4 \times 13} = 2\sqrt{13} \]
Now, we calculate the numeric value of \(2\sqrt{13}\).
- \(\sqrt{13} \approx 3.60555\)
Thus: \[ d \approx 2 \times 3.60555 \approx 7.2111 \]
Rounding to the nearest hundredth: \[ d \approx 7.21 \]
The length between the points \((7, 8)\) and \((3, 2)\) is approximately 7.21 units.