To find the length between the points \((-1, 6)\) and \( (5, -2)\), we 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:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates \((-1, 6)\) as \((x_1, y_1)\) and \((5, -2)\) as \((x_2, y_2)\):
- \(x_1 = -1\)
- \(y_1 = 6\)
- \(x_2 = 5\)
- \(y_2 = -2\)
Now we calculate the differences:
\[ x_2 - x_1 = 5 - (-1) = 6 \] \[ y_2 - y_1 = -2 - 6 = -8 \]
Next, we can substitute these values into the distance formula:
\[ d = \sqrt{(6)^2 + (-8)^2} \]
Calculating the squares:
\[ d = \sqrt{36 + 64} \] \[ d = \sqrt{100} \] \[ d = 10 \]
Therefore, the distance between the points \((-1, 6)\) and \( (5, -2)\) is \(10\).
The correct response is:
10