Write an equation in slope-intercept form for the perpendicular bisector of AB where A(-2,4) and B(4,6)
The Correct Answer and Explanation is:
To find the equation of the perpendicular bisector of the line segment connecting points A(-2,4) and B(4,6), follow these steps:
Step 1: Find the Midpoint of AB
The midpoint formula is: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)M=(2×1+x2,2y1+y2)
Substitute the coordinates of A(-2,4) and B(4,6): M=(−2+42,4+62)M=(22,102)M=(1,5)M = \left( \frac{-2 + 4}{2}, \frac{4 + 6}{2} \right) M = \left( \frac{2}{2}, \frac{10}{2} \right) M = (1, 5)M=(2−2+4,24+6)M=(22,210)M=(1,5)
So, the midpoint M(1,5)M(1, 5)M(1,5) is on the perpendicular bisector.
Step 2: Find the Slope of AB
The formula for the slope of a line through two points (x₁, y₁) and (x₂, y₂) is: m=y2−y1x2−x1m = \frac{y_2 – y_1}{x_2 – x_1}m=x2−x1y2−y1
Substitute the coordinates of A(-2, 4) and B(4, 6): m=6−44−(−2)=26=13m = \frac{6 – 4}{4 – (-2)} = \frac{2}{6} = \frac{1}{3}m=4−(−2)6−4=62=31
So, the slope of line AB is 13\frac{1}{3}31.
Step 3: Find the Slope of the Perpendicular Bisector
The slope of the perpendicular bisector is the negative reciprocal of the slope of AB. If the slope of AB is 13\frac{1}{3}31, then the slope of the perpendicular bisector will be: mperp=−3m_{\text{perp}} = -3mperp=−3
Step 4: Use the Point-Slope Form to Find the Equation of the Perpendicular Bisector
The point-slope form of a line is: y−y1=m(x−x1)y – y_1 = m(x – x_1)y−y1=m(x−x1)
We know the slope mperp=−3m_{\text{perp}} = -3mperp=−3 and the point on the line M(1,5)M(1, 5)M(1,5). Substituting these values into the point-slope form: y−5=−3(x−1)y – 5 = -3(x – 1)y−5=−3(x−1)
Simplify: y−5=−3x+3y – 5 = -3x + 3y−5=−3x+3
Add 5 to both sides: y=−3x+8y = -3x + 8y=−3x+8
Final Answer:
The equation of the perpendicular bisector of AB is: y=−3x+8y = -3x + 8y=−3x+8
This is the slope-intercept form of the perpendicular bisector
