Question

what is an equation of the perpendicular bisector of the line segment shown in the diagram below?task card 1

Explanation:

Step1: Find the mid - point of the line segment

The mid - point formula is \((\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\). Let the two endpoints be \((x_1,y_1)=(-5,-2)\) and \((x_2,y_2)=(3,2)\).

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The mid - point is \((-1,0)\).

Step2: Calculate the slope of the line segment

The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\).

$$ m=\frac{2-(-2)}{3-(-5)}=\frac{2 + 2}{3 + 5}=\frac{4}{8}=\frac{1}{2} $$

Step3: Determine the slope of the perpendicular bisector

The slope of a line perpendicular to a line with slope \(m\) is \(m_{\perp}=-\frac{1}{m}\). Since \(m = \frac{1}{2}\), then \(m_{\perp}=-2\).

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 \((x_0,y_0)=(-1,0)\) and \(m=-2\).

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Answer:

\(y=-2x - 2\)