A line segment has endpoints A(1, -5) and B(4, 1).
A. Determine the coordinates of two points, C and D that would make ABCD a parallelogram.
To make ABCD a parallelogram, we need to find points C and D that are parallel to line segment AB and have the same length as AB.
First, let's find the length and slope of line segment AB:
Length of AB = √((x₂ - x₁)² + (y₂ - y₁)²)
= √((4 - 1)² + (1 - (-5))²)
= √(3² + 6²)
= √(9 + 36)
= √45
Slope of AB = (y₂ - y₁) / (x₂ - x₁)
= (1 - (-5)) / (4 - 1)
= 6 / 3
= 2
Now, to find points C and D, we need to use the slope-intercept form y = mx + b or the point-slope form y - y₁ = m(x - x₁).
Since we already have a point on line segment AB, let's use the point-slope form:
For point C:
y - y₁ = m(x - x₁)
y - (-5) = 2(x - 1)
y + 5 = 2x - 2
y = 2x - 2 - 5
y = 2x - 7
The coordinates of point C are (x, y) = (x, 2x - 7).
Similarly, for point D:
y - y₁ = m(x - x₁)
y - (-5) = 2(x - 4)
y + 5 = 2x - 8
y = 2x - 8 - 5
y = 2x - 13
The coordinates of point D are (x, y) = (x, 2x - 13).
Therefore, the coordinates of points C and D that would make ABCD a parallelogram are:
C: (x, y) = (x, 2x - 7)
D: (x, y) = (x, 2x - 13)