Final answer:
The value of x for point E to make lines BC and DE parallel is approximately 6.619, which does not match any of the provided options.
Step-by-step explanation:
To determine the value of x for point E so that lines BC and DE are parallel, we must compare the slopes of BC and DE since parallel lines have equal slopes. The slope of a line through two points, (x1, y1) and (x2, y2), is given by the formula (y2 - y1) / (x2 - x1). Calculating the slope of BC using points B(1, 7.5) and C(11, -3), we get:
slope of BC = (-3 - 7.5) / (11 - 1) = -10.5 / 10 = -1.05
For DE to be parallel to BC, its slope must also be -1.05. Let's assume the x-coordinate of E is x. Using points D(-1, 2) and E(x, -6), the slope of DE will be:
slope of DE = (-6 - 2) / (x - (-1)) = -8 / (x + 1)
Setting this equal to the slope of BC and solving for x, we get:
-8 / (x + 1) = -1.05
8 = 1.05(x + 1)
8 = 1.05x + 1.05
8 - 1.05 = 1.05x
6.95 = 1.05x
x = 6.95 / 1.05
x = 6.619
None of the provided options (A) r = 3, (B) r = 2, (C) r = 5, or (D) r = 4 is correct as the value of x we found does not match any of them. It seems there may be a mistake in the question or the given options.