Final answer:
The equation for line k, the perpendicular bisector of PQ, in slope-intercept form is y = (-2/3)x + 4.
Step-by-step explanation:
To find the equation of line k which is the perpendicular bisector of PQ, we first need to find the midpoint of PQ. The midpoint formula is given by:
x = (x1 + x2) / 2
y = (y1 + y2) / 2
Substituting the values of P(10, 4) and Q(2, -8) into the formula, we get:
x = (10 + 2) / 2 = 6
y = (4 - 8) / 2 = -2
So, the midpoint of PQ is M(6, -2).
The slope of line PQ is given by:
mPQ = (y2 - y1) / (x2 - x1)
Substituting the values of P(10, 4) and Q(2, -8), we get:
mPQ = (-8 - 4) / (2 - 10) = -12 / -8 = 3/2
Since line k is the perpendicular bisector of PQ, it will have a negative reciprocal slope of -2/3. Using the point-slope form of a linear equation, where the slope is m and the point is (x1, y1), we get:
y - y1 = m(x - x1)
Substituting the values of m = -2/3 and (x1, y1) = (6, -2), we get:
y - (-2) = (-2/3)(x - 6)
y + 2 = (-2/3)(x - 6)
Rearranging the equation to slope-intercept form, y = mx + b, where b is the y-intercept, we get:
y = (-2/3)x + 4