Question

a line segment has endpoints at (4, -6) and (0, 2). what is the slope of the given line segment? what is the midpoint of the given line segment? what is the slope of the perpendicular bisector of the given line segment? what is the equation, in slope - intercept form, of the perpendicular bisector?

Explanation:

Step1: Calculate slope of line segment

$m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{2-( - 6)}{0 - 4}=-2$

Step2: Find mid - point

$(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})=(2,-2)$

Step3: Determine slope of perpendicular line

Since $m_1m_2=-1$ and $m_1=-2$, then $m_2=\frac{1}{2}$

Step4: Find y - intercept of perpendicular bisector

Substitute $x = 2,y=-2,m=\frac{1}{2}$ into $y=mx + b$: $-2=\frac{1}{2}\times2+b$, solve for $b=-3$

Answer:

1. For the slope of the given line segment:

• First, recall the slope - formula $m=\frac{y_2 - y_1}{x_2 - x_1}$. Let $(x_1,y_1)=(4, - 6)$ and $(x_2,y_2)=(0,2)$.
• Then $m=\frac{2-( - 6)}{0 - 4}=\frac{2 + 6}{-4}=\frac{8}{-4}=-2$.

1. For the mid - point of the given line segment:

• Recall the mid - point formula $(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})$.
• Substitute $x_1 = 4,y_1=-6,x_2 = 0,y_2 = 2$ into the formula: $(\frac{4+0}{2},\frac{-6 + 2}{2})=(2,-2)$.

1. For the slope of the perpendicular bisector:

• If two lines are perpendicular, the product of their slopes is $-1$. Let the slope of the given line be $m_1=-2$ and the slope of the perpendicular line be $m_2$. Then $m_1m_2=-1$.
• Solving for $m_2$, we get $m_2=\frac{1}{2}$.

1. For the equation of the perpendicular bisector in slope - intercept form ($y=mx + b$):

• We know the slope $m=\frac{1}{2}$ and the line passes through the mid - point $(2,-2)$.
• Substitute $x = 2,y=-2,m=\frac{1}{2}$ into $y=mx + b$: $-2=\frac{1}{2}\times2+b$.
• Simplify the right - hand side: $-2 = 1 + b$.
• Solve for $b$: $b=-3$.
• The equation of the perpendicular bisector is $y=\frac{1}{2}x-3$.