Bot GPT 4-o mini To find the length between the two points \((-1, 6)\) and \((5, -2)\), we can use the distance formula derived from the Pythagorean theorem. The formula for the distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
In this case, the points are:
- \((x_1, y_1) = (-1, 6)\)
- \((x_2, y_2) = (5, -2)\)
Now we can substitute these values into the formula:
\[ d = \sqrt{(5 - (-1))^2 + (-2 - 6)^2} \]
Calculating each part:
- \(5 - (-1) = 5 + 1 = 6\)
- \(-2 - 6 = -8\)
So we have:
\[ d = \sqrt{(6)^2 + (-8)^2} \]
Calculating the squares:
- \(6^2 = 36\)
- \((-8)^2 = 64\)
Now add these results:
\[ d = \sqrt{36 + 64} = \sqrt{100} \]
Finally, we take the square root:
\[ d = 10 \]
Thus, the length between the points \((-1, 6)\) and \((5, -2)\) is \(10\).
The correct response is 10.
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