To find the length between the points \((-1, 6)\) and \((5, 2)\), 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:
- \((x_1, y_1) = (-1, 6)\)
- \((x_2, y_2) = (5, 2)\)
Substituting in the values:
\[ d = \sqrt{(5 - (-1))^2 + (2 - 6)^2} \]
Calculating the differences:
\[ d = \sqrt{(5 + 1)^2 + (2 - 6)^2} = \sqrt{(6)^2 + (-4)^2} \]
Calculating the squares:
\[ d = \sqrt{36 + 16} = \sqrt{52} \]
Now simplifying \(\sqrt{52}\):
\[ d = \sqrt{4 \times 13} = 2\sqrt{13} \]
Now, we need to calculate \(2\sqrt{13}\) to get its decimal value. Using a calculator:
\(\sqrt{13} \approx 3.60555\)
Thus,
\[ d \approx 2 \times 3.60555 \approx 7.2111 \]
Rounding to the nearest hundredth gives us:
\[ d \approx 7.21 \]
This value does not match any of the given options exactly, but it seems like there might be a misunderstanding or a transcription error in the problem options since none of the provided distances (9.22, 100, 14, 10) are correct according to our calculation. Please double-check the provided options or the statements of the points.