Question

triangle abc has vertices a(0,0), b(6,8), and c(8,4). which equation represents the perpendicular bisector of bc?

Explanation:

Step1: Find the mid - point of \(BC\)

The mid - point formula for two points \((x_1,y_1)\) and \((x_2,y_2)\) is \((\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\).
For \(B(6,8)\) and \(C(8,4)\), the mid - point \(M\) is \((\frac{6 + 8}{2},\frac{8+4}{2})=(7,6)\).

Step2: Find the slope of \(BC\)

The slope formula for two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(m=\frac{y_2 - y_1}{x_2 - x_1}\).
For \(B(6,8)\) and \(C(8,4)\), the slope of \(BC\), \(m_{BC}=\frac{4 - 8}{8 - 6}=\frac{-4}{2}=-2\).

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 \(m\times m_{BC}=-1\) and \(m_{BC}=-2\), then \(m=\frac{1}{2}\).

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

The point - slope form of a line 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 point \(M(7,6)\) and \(m = \frac{1}{2}\), we have \(y - 6=\frac{1}{2}(x - 7)\).
Expand it: \(y-6=\frac{1}{2}x-\frac{7}{2}\).
\(y=\frac{1}{2}x-\frac{7}{2}+6\).
\(y=\frac{1}{2}x+\frac{5}{2}\).
Multiply through by \(2\) to get \(2y=x + 5\) or \(x-2y+5 = 0\).

Answer:

\(x-2y + 5=0\)