Question

What is the value of k if the coordinates of points A, B, and C are A(-1,-3), B(2,3), and C(6,k) and AB is perpendicular to BC?

a) 2
b) 5
c) -5
d) -2

Answer 1

The value of k is found by calculating the negative reciprocal of the slope of line AB and using it to set up the point-slope equation of line BC, which yields k = 5.

Step-by-step explanation:

To find the value of k when the coordinates of points A, B, and C are A(-1,-3), B(2,3), and C(6,k) and AB is perpendicular to BC, we must use the properties of perpendicular lines in the coordinate plane. The slope of line AB is given by the difference in y-coordinates divided by the difference in x-coordinates of points A and B:

mAB = (3 - (-3)) / (2 - (-1)) = 6 / 3 = 2.

The slope of any line perpendicular to AB will be the negative reciprocal of 2, which is -1/2. To find the value of k, use the point-slope form of the equation for the line BC:

y - y1 = m(x - x1).

Since we know point B (2,3) lies on BC and the slope m is -1/2:

3 - k = (-1/2)(6 - 2)

3 - k = -2

k = 3 + 2

k = 5.

Hence, the correct answer is (b) 5.