Answer:
C. y = ⁶/5x + ⁷/2
Step-by-step explanation:
First, find the slope of line AB that goes through A(3, 1) and B(-3, 6):
[tex] slope(m) = \frac{y_2 - y_1}{x_2 - x_1} = \frac{6 - 1}{-3 - 3} = \frac{5}{-6} [/tex].
Slope of line AB = -⅚.
The slope of the line that is a perpendicular bisector of line AB will be a negative reciprocal of the slope of line AB.
Thus:
Negative reciprocal of -⅚ = ⁶/5. (Reciprocal of ⅚, also the sign will change from positive to negative).
Next is to find the y-intercept, b, of the line.
To do this, you need to find the midpoint where the two lines intersect:
Therefore,
Midpoint (M) of AB, for A(3, 1) and B(-3, 6) is given as:
[tex] M(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}) [/tex]
Let [tex] A(3, 1) = (x_1, y_1) [/tex]
[tex] B(-3, 6) = (x_2, y_2) [/tex]
Thus:
[tex] M(\frac{3 +(-3)}{2}, \frac{1 + 6}{2}) [/tex]
[tex] M(\frac{0}{2}, \frac{7}{2}) [/tex]
[tex] M(0, \frac{7}{2}) [/tex]
Substitute x = 0, y = ⁷/2, and m = ⁶/5 into y = mx + b and find the value of b.
⁷/2 = ⁶/5(0) + b
⁷/2 = b
b = ⁷/2
The slope (m) and the y-intercept, b, of the line we are looking for are ⁶/5 and ⁷/2, respectively.
Therefore, substitute m = ⁶/5 and b = ⁷/2 into y = mx + b.
y = ⁶/5x + ⁷/2
The equation that is the perpendicular bisector of the line AB is y = ⁶/5x + ⁷/2.
