Answer:
To find the location of point L, we need to use the fact that point K lies on the perpendicular bisector of segment JL.
The perpendicular bisector of a segment is a line that passes through the midpoint of the segment and is perpendicular to the segment.
First, we can find the midpoint of segment JL:
Midpoint of JL = ((-5 + x)/2, (-8 + y)/2)
where (x, y) is the location of point L.
Since point K lies on the perpendicular bisector of JL, the line passing through K and the midpoint of JL will be perpendicular to JL.
The slope of the line passing through K and the midpoint of JL is:
m = (2 - (-8))/(-9 - (-5 + x)/2) = 10/(-9 - (5 - x)/2) = -4/(x - 7)
where we have used the fact that the line passing through K and the midpoint of JL is perpendicular to JL, so its slope is the negative reciprocal of the slope of JL.
Now, we can use the fact that the line passing through K and the midpoint of JL contains point K(-9, 2) to write the equation of the line:
y - 2 = m(x + 9)
Substituting the expression we found for m, we get:
y - 2 = (-4/(x - 7))(x + 9)
Simplifying this equation, we get:
y = -4x/ (x - 7) - 2
Now, we can substitute the x-coordinates of the answer choices to see which one gives us a valid y-coordinate:
A. (1, -2): y = -4(1)/(-6) - 2 = -1 --> Incorrect
B. (-1, -12): y = -4(-1)/(-8) - 2 = 1/2 --> Incorrect
C. (-3, -2): y = -4(-3)/(-10) - 2 = 6/5 --> Incorrect
D. (-3, -12): y = -4(-3)/(-10 - 4) - 2 = -26/7 --> Correct
Therefore, the location of point L is (-3, -12), which is answer choice D.