Question

question
find an equation for the perpendicular bisector of the line segment whose endpoints are
$(9,6)$ and $(1,-4)$.
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Explanation:

Step1: Find the mid - point

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 the points \((9,6)\) and \((1,-4)\), we have \(x=\frac{9 + 1}{2}=5\) and \(y=\frac{6+( - 4)}{2}=1\). So the mid - point is \((5,1)\).

Step2: Find the slope of the line segment

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 the points \((9,6)\) and \((1,-4)\), \(m=\frac{-4 - 6}{1 - 9}=\frac{-10}{-8}=\frac{5}{4}\).

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 line segment be \(m_1=\frac{5}{4}\) and the slope of the perpendicular bisector be \(m_2\). Then \(m_1\times m_2=-1\), so \(m_2=-\frac{4}{5}\).

Step4: Use the point - slope form

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 mid - point \((5,1)\) as \((x_0,y_0)\) and \(m =-\frac{4}{5}\), we get \(y - 1=-\frac{4}{5}(x - 5)\).
Expand it: \(y-1=-\frac{4}{5}x + 4\), then \(y=-\frac{4}{5}x+5\) or \(4x + 5y=25\).

Answer:

\(4x + 5y=25\)