Question

1. which equation represents the perpendicular bisector of \\( \overline { a b } \\) whose endpoints are \\( a ( 8,2 ) \\) and \\( b ( 0,6 ) \\) ?

a) \\( y = 2 x - 4 \\)
b) \\( y = - \frac { 1 } { 2 } x + 2 \\)
c) \\( y = - \frac { 1 } { 2 } x + 6 \\)
d) \\( y = 2 x - 12 \\)

Explanation:

Step1: Find the mid - point of \(\overline{AB}\)

The mid - point formula is \(M(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\).
For \(A(8,2)\) and \(B(0,6)\), \(x_1 = 8,y_1 = 2,x_2 = 0,y_2 = 6\).
\(M(\frac{8 + 0}{2},\frac{2+6}{2})=M(4,4)\)

Step2: Find the slope of \(\overline{AB}\)

The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\).
\(m_{AB}=\frac{6 - 2}{0 - 8}=\frac{4}{-8}=-\frac{1}{2}\)

Step3: Find the slope of the perpendicular bisector

If two lines are perpendicular, \(m_1\times m_2=-1\). Let \(m_1 =-\frac{1}{2}\), then \(m_2 = 2\)

Step4: Use the point - slope form \(y - y_0=m(x - x_0)\)

Using the mid - point \((x_0,y_0)=(4,4)\) and \(m = 2\)
\(y - 4=2(x - 4)\)
\(y-4=2x - 8\)
\(y=2x-4\)

Answer:

A. \(y = 2x-4\)