Question

Which of the following equations is the perpendicular bisector of the line AB given:
A(3,1) & B(-3,6)
O y = 6/5x + 4
O y = 6/5x + 9/2
O y = 6/5x + 7/2
O y = -5/6x + 7/2

Answer 1

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):

$slope(m) = \frac{y_2 - y_1}{x_2 - x_1} = \frac{6 - 1}{-3 - 3} = \frac{5}{-6}$.

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:

$M(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2})$

Let $A(3, 1) = (x_1, y_1)$

$B(-3, 6) = (x_2, y_2)$

Thus:

$M(\frac{3 +(-3)}{2}, \frac{1 + 6}{2})$

$M(\frac{0}{2}, \frac{7}{2})$

$M(0, \frac{7}{2})$

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.