y - 5 = -x + 4, or
y = -x + 9. This line is the perpendicular bisector of AB.
Since (7,t) lies on this line, substituting 7 for x must yield t as the value of y:
t = -7 + 9, or
t = 2. C is then the point (7,2).
Now we have a line between (4,5) and (7,2), that is, between the midpoint of AB and the point C. Since AB bisects CD, then CD must extend beyond AB by an amount equal to the distance from M to C. We need to find that distance, then double it and find the coordinates of D, the endpoint of CD.
Use the distance formula: d = √((x₂-x₁)² + (y₂-y₁)²).
d = √((7-4)² + (2-5)²)
= √(3² + (-3)²)
= √(9 + 9)
= √18
= 3√2.
The total distance is twice this, or 6√2. Put this into the distance formula, along with the coordinates of C, to find x and y, the coordinates of D.
Since the point is on the line y = -x + 9, substitute (-x+9) for y:
6√2 = √((x-7)² + ((-x+9)-2)²), or
(6√2)² = (x-7)² + ((-x+9)-2)², or
72 = (x-7)² + (-x+7)², or
x² - 14x + 49 + x² - 14x + 49 = 72, or
2x² - 28x + 26 = 0, or
x² - 14x + 13 = 0, or
(x-1)(x-13) = 0, so
x = 1 or x = 13.
x = 13 corresponds to the point on the "wrong" side of C: it is 6√2 units away, but in the opposite direction from the point M (toward AB).
Use x = 1 instead. y = -x + 9, so y = 8, and point D is (1,8).