Final answer:
Given points D(3,1), E(7,2), and F(4,5), we can find the midpoint of segment DE and determine the slope of DE. Using the negative reciprocal of the slope, we can find the equation of the perpendicular bisector of DE. By plugging in the coordinates of point F into the equation of the perpendicular bisector, we can determine if F lies on the perpendicular bisector of DE.
Step-by-step explanation:
Given points D(3,1), E(7,2), and F(4,5), we can find the midpoint of segment DE by finding the average of the x-coordinates and the average of the y-coordinates. The midpoint of DE is ((3+7)/2, (1+2)/2) = (5,1.5).
Next, we find the slope of DE by using the formula: slope = (y2-y1)/(x2-x1). The slope of DE is (2-1)/(7-3) = 1/4. Since the perpendicular bisector of a line segment has a slope that is the negative reciprocal of the original line, the slope of the perpendicular bisector will be -4.
Now, we can check if point F lies on the perpendicular bisector of DE. We find the equation of the perpendicular bisector using the midpoint coordinates, (5,1.5), and the slope, -4. The equation is y-1.5 = -4(x-5). Plugging in the coordinates of point F, we get 5-1.5 = -4(4-5), which simplifies to 3.5 = 4. Since this equation is not satisfied, we can conclude that F does not lie on the perpendicular bisector of DE.