A line segment has endpoints at \( (-4,5) \) and \( (4,-11) \). A point on the segment has a distance that is \( \frac{1}{4} \) of the length of the segment from endpoint \( (-4,5) \). What are the coordinates of the point? Point:

To find the coordinates of the point on the segment that is \( \frac{1}{4} \) of the way from endpoint \( (-4,5) \) to endpoint \( (4,-11) \), we first calculate the length of the segment. The length of the segment, using the distance formula, is given by: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{(4 - (-4))^2 + (-11 - 5)^2} = \sqrt{(4 + 4)^2 + (-16)^2} = \sqrt{8^2 + 16^2} = \sqrt{64 + 256} = \sqrt{320} = 8\sqrt{5} \] Since we need to find a point that is \( \frac{1}{4} \) of the way along the segment, we calculate: \[ \text{Distance from } (-4,5) = \frac{1}{4} \times 8\sqrt{5} = 2\sqrt{5} \] Next, to find the coordinates of the point, we first find the direction vector of the segment by subtracting the coordinates of the endpoints: \[ \text{Direction vector} = (4 - (-4), -11 - 5) = (8, -16) \] Now, we can use this direction vector to determine the coordinates of the point. The point that is \( \frac{1}{4} \) of the way from \( (-4, 5) \) can be expressed as: \[ \text{Point} = \left( -4, 5 \right) + \frac{2\sqrt{5}}{8\sqrt{5}} \cdot (8, -16) = \left( -4, 5 \right) + \frac{1}{4} \cdot (8, -16) = \left( -4 + 2, 5 - 4 \right) = \left( -2, 1 \right) \] Therefore, the coordinates of the point are \( (-2, 1) \).