To find the distance between the two points \((7, 8)\) and \((3, 2)\), we can use the distance formula, which is derived from the Pythagorean Theorem. The distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
In this case, we have:
- \(x_1 = 7\), \(y_1 = 8\)
- \(x_2 = 3\), \(y_2 = 2\)
Now we can substitute these values into the formula:
- Calculate \(x_2 - x_1\) and \(y_2 - y_1\):
\[ x_2 - x_1 = 3 - 7 = -4 \] \[ y_2 - y_1 = 2 - 8 = -6 \]
- Substitute these differences into the distance formula:
\[ d = \sqrt{(-4)^2 + (-6)^2} \]
- Calculate the squares:
\[ d = \sqrt{16 + 36} \]
- Add the squares:
\[ d = \sqrt{52} \]
- Simplify \(\sqrt{52}\):
\[ d = \sqrt{4 \times 13} = 2\sqrt{13} \]
- Now, calculate the numerical value of \(2\sqrt{13}\):
Using a calculator, \(\sqrt{13} \approx 3.60555\), so:
\[ d \approx 2 \times 3.60555 \approx 7.2111 \]
- Finally, round to the nearest hundredth:
\[ d \approx 7.21 \]
Thus, the length between the two points is approximately 7.21.