Final answer:
To calculate the possible values of k, use the distance formula to find the lengths of AB and BC. Set the lengths equal to each other to solve for k. the coordinates of D are (-2, 0).
Step-by-step explanation:
To calculate the possible values of k, we need to determine the length of AB and BC. The distance formula can be used to find the lengths. AB = √[(x2 - x1)^2 + (y2 - y1)^2] = √[(5 - (-3))^2 + (1 - 7)^2] = √[(8)^2 + (-6)^2] = 10.
Since AB = BC, BC also has a length of 10. Using the distance formula again, we get 10 = √[(5 - (-1))^2 + (1 - k)^2].
Simplifying the equation, we have 10 = √[(6)^2 + (1 - k)^2]. Squaring both sides, we get 100 = (36) + (1 - k)^2. 64 = (1 - k)^2. Taking the square root, we have 8 = 1 - k or 8 = -(1 - k). Therefore, k can be either -7 or 9.
To calculate the coordinates of D, we need to find the midpoint of AB and the slope of AB. The midpoint is ((x1 + x2)/2, (y1 + y2)/2), which in this case is ((-3 + 5)/2, (7 + 1)/2) = (1, 4).
The slope of AB is (y2 - y1)/(x2 - x1) = (1 - 7)/(5 - (-3)) = -6/8 = -3/4.
The perpendicular bisector of AB has a slope that is the negative reciprocal of the slope of AB, which is 4/3.
Using the point-slope form of a line y - y1 = m(x - x1), we can substitute the values into the equation y - 4 = (4/3)(x - 1).
Setting y = 0, we can solve for x to find the x-coordinate of D. 0 - 4 = (4/3)(x - 1). -4 = (4/3)(x - 1). Multiplying both sides by 3 to eliminate the fraction, we get -12 = 4(x - 1). -12 = 4x - 4. Adding 4 to both sides, we have -8 = 4x. Dividing both sides by 4, we get x = -2.
Therefore, the coordinates of D are (-2, 0).