Final answer:
Two points on the perpendicular bisector of AB for points A = (-2, 7) and B = (3, 3) can be found by calculating the midpoint and the slope of AB and then deriving the equation of the bisector. Any point P on the perpendicular bisector will have equal segments PA and PB, exemplified by two calculated points P1 (2, 8.875) and P2 (-1, 3.125).
Step-by-step explanation:
To find two points P that are on the perpendicular bisector of AB for points A = (-2, 7) and B = (3, 3), we first need to find the midpoint of segment AB as well as its slope to determine the slope of the perpendicular bisector.
The midpoint M of AB is calculated as M = ((x1+x2)/2, (y1+y2)/2). Therefore, M = ((-2+3)/2, (7+3)/2) = (0.5, 5).
The slope of AB, mAB, is (y2-y1)/(x2-x1), which gives mAB = (3-7)/(3+2) = -4/5. The slope of the perpendicular bisector, mperp, is the negative reciprocal of mAB, so mperp = 5/4.
Using the point-slope form of a line, y - y1 = m(x - x1), with point M and slope mperp, the equation of the perpendicular bisector is y - 5 = (5/4)(x - 0.5).
Now that we have the equation, we can pick any x-values to find corresponding y-values for points on the perpendicular bisector. For example, for x = 2, we have y - 5 = (5/4)(2 - 0.5), which gives point P1 (2, 8.875). For x = -1, we have y - 5 = (5/4)(-1 - 0.5), which gives point P2 (-1, 3.125).
Segments PA and PB will be equal for any point P on the perpendicular bisector because by definition, a perpendicular bisector crosses a segment at its midpoint and at a right angle, ensuring that any point on it is equidistant from the endpoints of the original segment.