Question

Point A has coordinates (-24, -54)
Point B has coordinates (40, -46)
Find the equation of the perpendicular bisector of line AB.
ANSWER ASAP

Answer 1

Answer:

$y=-8x+14$

Step-by-step explanation:

Hi there!

What we need to know:

• A perpendicular bisector of a line segment is 1) perpendicular to the line segment and 2) passes through the midpoint of the line segment
• Perpendicular lines always have slopes that are negative reciprocals (ex. -2 and 1/2)
• Linear equations are typically organized in slope-intercept form: $y=mx+b$ where m is the slope and b is the y-intercept (the value of  when x is 0)

1) Determine the midpoint of the line segment

Midpoint: $(\frac{x_1+x_2}{2} ,\frac{y_1+y_2}{2} )$ where the coordinates of the endpoints are $(x_1,y_1)$ and $(x_2,y_2)$

Plug in the endpoints (-24, -54) and (40, -46)

$(\frac{-24+40}{2} ,\frac{-54+(-46)}{2} )\\(\frac{-24+40}{2} ,\frac{-54-46}{2} )\\(\frac{16}{2} ,\frac{-100}{2} )\\(8 ,-50)$

Therefore, the midpoint of line AB is (8,-50).

2) Determine the slope of the line segment

This will help us find the equation of the perpendicular bisector.

slope = $\frac{y_2-y_1}{x_2-x_1}$ where two given points are $(x_1,y_1)$ and $(x_2,y_2)$

Plug in the endpoints (-24, -54) and (40, -46)

$= \frac{-46-(-54)}{40-(-24)}\\= \frac{-46+54}{40+24}\\= \frac{8}{64}\\= \frac{1}{8}$

Therefore, the slope of line AB is $\frac{1}{8}$.

3) Determine the slope of the perpendicular bisector

Because perpendicular lines always have slopes that are negative reciprocals, the slope of the perpendicular bisector is -8 (the negative reciprocal of 1/8). Plug this slope into $y=mx+b$:

$y=-8x+b$

4) Determine the y-intercept (b) of the perpendicular bisector

$y=-8x+b$

Recall that we found the midpoint of line AB, (8,-50). The perpendicular bisector passes through this point. Plug (8,-50) into $y=-8x+b$ and solve for b:

$-50=-8(8)+b\\-50=-64+b$

Add 64 to both sides to isolate b

$-50+64=-64+b+64\\14=b$

Therefore, the y-intercept of the line is 14. Plug this back into $y=-8x+b$:

$y=-8x+14$

I hope this helps!