Question

A perpendicular bisector, CĎ, is drawn through point C on line AB. If the coordinates of point A are (3, 2) and the coordinates of point B are (7, 6), the x-intercept of CĎ is:

A) (3.0)
B) (18, 0)
C) (9, 0)
D) (45/2, 0)

Answer 1

To find the x-intercept of the perpendicular bisector CD, we first need to find the midpoint of line AB, which is the point C. The equation of the perpendicular bisector is y = -x + 9. To find the x-intercept, we set y = 0 and solve for x. The x-intercept of CD is (9, 0), so the answer is option C.

Step-by-step explanation:

To find the x-intercept of the perpendicular bisector CD, we first need to find the midpoint of line AB, which is the point C. The midpoint formula is given by: C = [(x1 + x2)/2, (y1 + y2)/2]. Substituting the values of A (3, 2) and B (7, 6), we get C = [(3 + 7)/2, (2 + 6)/2] = [(10/2), (8/2)] = [5, 4].

Next, we need to find the slope of line AB. The slope formula is given by: m = (y2 - y1)/(x2 - x1). Substituting the values of A (3, 2) and B (7, 6), we get m = (6 - 2)/(7 - 3) = 4/4 = 1.

Since the perpendicular bisector is perpendicular to AB, its slope will be the negative reciprocal of 1, which is -1. Using the slope-intercept form of a line, y = mx + b, we can substitute the values of C (5, 4) and -1 for m to find the equation of the perpendicular bisector. Solving for b, we get: 4 = -1(5) + b, which gives b = 9.

The equation of the perpendicular bisector is therefore y = -x + 9. To find the x-intercept, we set y = 0 and solve for x. 0 = -x + 9, which gives x = 9. Therefore, the x-intercept of CD is (9, 0), so the answer is option C.