P (5,) and Q (-1, 2) are points on a straight line. Find the equation of the perpendicular bisector of PQ: y = mx+c
how to find missing
If the y-coordinate of P is missing, you need to supply it.
If it is not given, then use w to complete the problem.
P(5,w),Q(-1,2),
Mid-point: M((5-1)/2, (w+2)/2)=(Mx,My)
Slope of PQ:
m=(2-w)/(-1-5)
slope of perpendicular:
m'=-(-1-5)/(2-w)=6/(2-w)
Perpendicular bisector must pass through M, use the Point-slope form:
(y-y1)=m(x-x1)
so
y-My = (6/(2-w)) (x-Mx)
To find the equation of the perpendicular bisector of line PQ, you need to find the slope (m) and the y-intercept (c).
Step 1: Find the midpoint of line PQ:
The midpoint (M) of line PQ can be found using the midpoint formula:
M = [(x₁ + x₂) / 2, (y₁ + y₂) / 2]
Given the points P(5, ) and Q(-1, 2), the midpoint is:
M = [(5 + (-1)) / 2, ( + 2) / 2]
M = [2, ]
Step 2: Find the slope of line PQ:
The slope of line PQ (m₁) can be found using the slope formula:
m₁ = (y₂ - y₁) / (x₂ - x₁)
Given the points P(5, ) and Q(-1, 2), the slope is:
m₁ = (2 - ) / (-1 - 5)
m₁ = (2 - ) / (-6)
Step 3: Find the negative reciprocal of the slope (m) to get the slope of the perpendicular bisector:
The slope of the perpendicular bisector (m₂) is the negative reciprocal of m₁:
m₂ = -1 / m₁
Step 4: Use the point-slope form to find the equation of the perpendicular bisector:
The point-slope form of a line is: y - y₁ = m(x - x₁)
Using point M(2, ) and the slope m₂, the equation of the perpendicular bisector is:
y - = m₂(x - 2)
Now you can substitute the value of slope m₂ in the equation to get the final answer.