A perpendicular bisector divides a line segment into two equal halves at a 90∘ angle.
Step 1: Find the midpoint of the line segment pq
The perpendicular bisector passes through the midpoint M of points p(1,4) and q(k,3).
M=(21+k,24+3)=(2k+1,27)
Step 2: Find the slope of the line segment pq
The slope (m1) of the line segment joining p(1,4) and q(k,3) is given by:
m1=k−13−4=k−1−1
Step 3: Find the slope of the perpendicular bisector
Let the slope of the perpendicular bisector be m2. Since the lines are perpendicular to each other:
m1⋅m2=−1
(k−1−1)⋅m2=−1
m2=k−1
Step 4: Form the equation of the perpendicular bisector
We know the perpendicular bisector has a slope m2=k−1 and a y-intercept of −4. The point-slope form or slope-intercept form (y=mx+c) gives:
y=(k−1)x−4— (Equation 1)
Step 5: Substitute the midpoint into the line equation
Since the midpoint M(2k+1,27) lies directly on the perpendicular bisector, substitute its coordinates into Equation 1:
27=(k−1)(2k+1)−4
Multiply the entire equation by 2 to eliminate the denominators:
7=(k−1)(k+1)−8
Using the algebraic identity (k−1)(k+1)=k2−1:
7=k2−1−8
7=k2−9
Rearrange the terms to solve for k2:
k2=7+9
k2=16
k=±16
k=±4
Thus, the possible values of k are −4 and 4.
Correct Answer: A) −4 and 4
