Respuesta :
Perpendicular means that the slope of one line is the opposite reciprocal of the other; bisector means that the line that is perpendicular is cutting the other one exactly in half. So we have to find 2 things about BC: the first is the midpoint which will then serve as the x and y coordinates in our new equation, and second is the slope of it. First things first: the midpoint formula is this: [tex]M=( \frac{ x_{1}+ x_{2} }{2} , \frac{ y_{1} + y_{2} }{2} )[/tex]. Using our coordinates we fill in that formula like this: [tex]M=( \frac{6+8}{2}, \frac{8+4}{2} ) [/tex] which of course gives us a midpoint of (7, 6). Now for the slope of the line BC: [tex]m= \frac{4-8}{8-6} =-2[/tex]. But since we need the perpendicular slope of that we will take its opposite reciprocal which is 1/2. Now use the x and y coordinates of the midpoint and that newly found slope and write your equation to solve for b. [tex]6= \frac{1}{2}(7)+b [/tex]. Solving that for b gives you that b = 5/2. So the equation of the line that is the perpendicular bisector of BC (drum roll, please...) is [tex]y= \frac{1}{2} x+ \frac{5}{2} [/tex]
The equation of the perpendicular bisector of line BC is :
[tex]y=\frac{1}{2}x+\frac{5}{2}[/tex]
The given coordinates are:
A(0, 0), B(6, 8), C(8, 4)
Find the midpoint of line BC
[tex]x=\frac{x_1+x_2}{2} \\\\x=\frac{6+8}{2} \\\\x=7\\\\\\y=\frac{y_1+y_2}{2} \\\\y=\frac{8+4}{2} \\\\y=6[/tex]
The coordinate of the midpoint of line BC is (7, 6)
Find the slope of line BC
[tex]m=\frac{y_2-y_1}{x_2-x_1}\\\\m=\frac{4-8}{8-6} \\\\m=-2[/tex]
The equation of the perpendicular bisector of BC is:
[tex]y=\frac{-1}{m}x+b\\\\[/tex]
Solve for the y-intercept, b
[tex]6=\frac{-1}{-2}(7)+b \\\\12=7+2b\\\\2b=12-7\\\\b=5/2[/tex]
The equation of the perpendicular bisector of line BC is therefore:
[tex]y=\frac{1}{2}x+\frac{5}{2}[/tex]
Learn more on perpendicular bisector here: https://brainly.com/question/11006922
