Question

find the equation of the perpendicular bisector of:
a a(5,2) and b(1,4) b a(-1,5) and b(5,3)
c m(7,2) and n(-1,6) d o(0,0) and p(9,0)

Explanation:

Step1: Find the mid - point of the line segment

The mid - point formula is \((\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\).
For points \(M(7,2)\) and \(N(-1,6)\), the mid - point \(C\) is \((\frac{7+( - 1)}{2},\frac{2 + 6}{2})=(3,4)\).

Step2: Find the slope of the line segment \(MN\)

The slope formula is \(m=\frac{y_2-y_1}{x_2-x_1}\).
For points \(M(7,2)\) and \(N(-1,6)\), \(m_{MN}=\frac{6 - 2}{-1-7}=\frac{4}{-8}=-\frac{1}{2}\).

Step3: Find the slope of the perpendicular bisector

If two lines are perpendicular, the product of their slopes is \(-1\). Let the slope of the perpendicular bisector be \(m\).
Since \(m\times m_{MN}=-1\) and \(m_{MN}=-\frac{1}{2}\), then \(m = 2\).

Step4: Use the point - slope form to find the equation of the perpendicular bisector

The point - slope form is \(y - y_0=m(x - x_0)\), where \((x_0,y_0)\) is a point on the line and \(m\) is the slope.
Using the mid - point \((3,4)\) and \(m = 2\), we have \(y-4=2(x - 3)\).
Expand it: \(y-4=2x-6\).
Rearrange to the general form: \(2x-y - 2=0\).

Answer:

The equation of the perpendicular bisector of the line segment joining \(M(7,2)\) and \(N(-1,6)\) is \(2x-y - 2 = 0\).