To solve these questions, we first need to find the length of the line segment with endpoints (4, 7) and (1, 11) using the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Here, \( (x_1, y_1) = (4, 7) \) and \( (x_2, y_2) = (1, 11) \).
Substituting these values into the formula:
\[ d = \sqrt{(1 - 4)^2 + (11 - 7)^2} \] \[ = \sqrt{(-3)^2 + (4)^2} \] \[ = \sqrt{9 + 16} \] \[ = \sqrt{25} \] \[ = 5 \]
So, the length of the line segment is 5.
Now, we need to determine which of the given line segments has the same length. We will also compute the lengths of the other line segments provided in the options using the distance formula.
Option A:
Endpoints: \((-6, 4)\) and \((2, -5)\) \[ d_A = \sqrt{(2 - (-6))^2 + (-5 - 4)^2} = \sqrt{(8)^2 + (-9)^2} = \sqrt{64 + 81} = \sqrt{145} \approx 12.04 \]
Option B:
Endpoints: \((1, -4)\) and \((9, 2)\) \[ d_B = \sqrt{(9 - 1)^2 + (2 - (-4))^2} = \sqrt{(8)^2 + (6)^2} = \sqrt{64 + 36} = \sqrt{100} = 10 \]
Option C:
Endpoints: \((-3, 1)\) and \((4, 5)\) \[ d_C = \sqrt{(4 - (-3))^2 + (5 - 1)^2} = \sqrt{(7)^2 + (4)^2} = \sqrt{49 + 16} = \sqrt{65} \approx 8.06 \]
Option D:
Endpoints: \((5, 3)\) and \((1, 6)\) \[ d_D = \sqrt{(1 - 5)^2 + (6 - 3)^2} = \sqrt{(-4)^2 + (3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \]
Conclusion:
- The length of the original line segment is 5.
- The line segments that also has a length of 5 is Option D: A line segment with endpoints (5, 3) and (1, 6).
Thus, the final answers are:
- Question 1: The length of the segment is 5 (though the options do not include this, they seem to be incorrect).
- Question 2: The answer is D (A line segment with endpoints (5, 3) and (1, 6)).