Question

select all equations that represent the perpendicular bisector of the line segment with endpoints at (1, 5) and (9, 1).
□ a) ( y = 2x - 7 )
□ b) ( y - 3 = 2(x - 5) )
□ c) ( y - 3 = -2(x - 5) )
□ d) ( y - 3 = -\frac{1}{2}(x - 5) )
□ e) ( y = -2x + 13 )
□ f) ( y - 3 = \frac{1}{2}(x - 5) )

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 the points \((1,5)\) and \((9,1)\), the mid - point is \((\frac{1 + 9}{2},\frac{5+1}{2})=(5,3)\).

Step2: Find the slope of the line segment

The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\).
For the points \((1,5)\) and \((9,1)\), \(m=\frac{1 - 5}{9 - 1}=\frac{-4}{8}=-\frac{1}{2}\).

Step3: Find the slope of the perpendicular bisector

The slope of a perpendicular line to a line with slope \(m\) is \(m'=-\frac{1}{m}\).
Since \(m =-\frac{1}{2}\), the slope of the perpendicular bisector \(m' = 2\).

Step4: Write the equation of the perpendicular bisector in point - slope form

The point - slope form is \(y - y_0=m'(x - x_0)\), where \((x_0,y_0)=(5,3)\) and \(m' = 2\).
So \(y - 3=2(x - 5)\).

Step5: Convert the point - slope form to slope - intercept form

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Answer:

A. \(y = 2x-7\), B. \(y - 3=2(x - 5)\)