Final answer:
To find the equation of the line that is the perpendicular bisector of segment PQ, calculate the midpoint M(6,-2), find the negative reciprocal of the slope of PQ which is -2/3, and use the point-slope form to determine the equation y = -2/3x + 6.
Step-by-step explanation:
The student is asking how to find the equation of a line that is the perpendicular bisector of line segment PQ, with given endpoints P(10,4) and Q(2,-8). First, we need to find the midpoint of PQ, which is also a point on the perpendicular bisector. The midpoint M can be found by averaging the x and y coordinates of P and Q, resulting in M(6,-2).
Next, we calculate the slope of PQ. The slope of line segment PQ is the change in y divided by the change in x, which gives us (4 - (-8)) / (10 - 2) = 12/8 = 1.5. The slope of the perpendicular bisector is the negative reciprocal of this slope, so it is -2/3.
Using the point-slope form y - y1 = m(x - x1), where m is the slope and (x1, y1) is a point on the line (the midpoint in our case), we can write the equation of the perpendicular bisector as y - (-2) = -2/3(x - 6). Simplifying, we get y = -2/3x + 6.