Final answer:
The perpendicular bisector L of segment AB has a slope of -3 and the equation y = -3x - 7.
Step-by-step explanation:
To find the equation of the line L that is the perpendicular bisector of segment AB, we first need to determine the midpoint of AB and the slope of AB. The midpoint can be found by averaging the x-coordinates and the y-coordinates of points A and B separately.
The midpoint M of AB is given by:
M = ((-10 + 2)/2, (3 + 7)/2)
M = (-4, 5).
The slope of AB is calculated by the formula Δy/Δx where Δy is the change in the y-coordinates and Δx is the change in the x-coordinates of points A and B. So, the slope of AB is:
Δy = 7 - 3 = 4
Δx = 2 - (-10) = 12
Slope of AB = Δy/Δx = 4/12 = 1/3.
The slope of any line perpendicular to AB would be the negative reciprocal of the slope of AB. Therefore, the slope of L is:
Slope of L = -1/(1/3) = -3.
Now, using the midpoint M as a point on L and the slope we just found, we can use the point-slope form of the equation of a line to find the equation of L.
y - y1 = m(x - x1)
y - 5 = -3(x - (-4))
y - 5 = -3(x + 4).
Now converting the point-slope form to the slope-intercept form y = mx + b, we get:
y = -3x - 12 + 5
y = -3x - 7.
Therefore, the slope of line L is -3 and its equation is y = -3x - 7 in slope-intercept form.