The equation of the perpendicular bisector of the line segment
joining the points (1 , 2) and (7 , 4) is y = -3x + 15
Step-by-step explanation:
Let us revise some rules
- The product of the slopes of the perpendicular lines is -1, that means one of them is additive and multiplicative inverse to the other. If the slope of one is m, then the slope of the other is
![-\frac{1}{m}](https://tex.z-dn.net/?f=-%5Cfrac%7B1%7D%7Bm%7D)
- The midpoint of a segment whose endpoints are
and
is ![(\frac{x_{1}+x_{2}}{2},\frac{y_{1}+y_{2}}{2})](https://tex.z-dn.net/?f=%28%5Cfrac%7Bx_%7B1%7D%2Bx_%7B2%7D%7D%7B2%7D%2C%5Cfrac%7By_%7B1%7D%2By_%7B2%7D%7D%7B2%7D%29)
- The slope of a line that passes through points
and
is ![m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}](https://tex.z-dn.net/?f=m%3D%5Cfrac%7By_%7B2%7D-y_%7B1%7D%7D%7Bx_%7B2%7D-x_%7B1%7D%7D)
∵ A line passes through points (1 , 2) and (7 , 4)
∴
= 1 and
= 7
∴
= 2 and
= 4
∴ The slope of the line = ![\frac{4-2}{7-1}=\frac{2}{6}=\frac{1}{3}](https://tex.z-dn.net/?f=%5Cfrac%7B4-2%7D%7B7-1%7D%3D%5Cfrac%7B2%7D%7B6%7D%3D%5Cfrac%7B1%7D%7B3%7D)
To find the slope of the perpendicular bisector of this line reciprocal
its slope and change its sign
∵ The reciprocal of
is 3
∴ The slope of the perpendicular bisector = -3
Let us find the mid point of the line
∵
= 1 and
= 7
∵
= 2 and
= 4
∴ The midpoint =
= (4 , 3)
∵ The form of the equation is y = mx + b, where m is the slope of
the line and b is the y-intercept
∵ m = -3
∴ The equation of the line is y = -3x + b
- To find b substitute x and y in the equation by the coordinates
of the midpoint
∵ The coordinates of the midpoint are x = 4 and y = 3
∴ 3 = -3(4) + b
∴ 3 = -12 + b
- Add 12 to both sides
∴ 15 = b
∴ the equation of the perpendicular bisector is y = -3x + 15
The equation of the perpendicular bisector of the line segment
joining the points (1 , 2) and (7 , 4) is y = -3x + 15
Learn more:
You can learn more about the perpendicular lines in brainly.com/question/2601054
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