Point A has coordinates (3,2) and point B has coordinates (9,-4). Find, by calculation, which one of the points C(4,-3), D(10,3) or E(5,-1) does not lie on the perpendicular bisector of AB. Please show all working and explanations. Medal will be awarded.

Question

whpalmer4 · Answer

Use the slope formula to find the slope of the line between A and B:

$$m = \frac{y_2 - y_1}{x_2-x_1}$$Now use the point-slope formula to write the equation of the line from A to B:
$$y-y_1 = m(x-x_1)$$You can use either A or B as $(x_1,y_1)$

Next, find the slope of the perpendicular bisector of AB, which will be the negative reciprocal of the slope you found: $$m_{bisector} = -\frac{1}{m}$$

Find the midpoint of AB by using the fact that the midpoint of a line segment is located at the average of the x coordinates and the y coordinates.

Use the slope of the bisector and the coordinates of the midpoint and the point-slope formula above to find the equation of the perpendicular bisector.

Now substitute each point into that equation and see which one doesn't produce a true number sentence.