Martin S. answered • 04/29/21
Patient, Relaxed PhD Molecular Biologist for Science and Math Tutoring
You can use the point slope formula to find the equation for the line in question. Since the line is perpendicular to the line segment, the slope of the line segment times the slope of the unknown line is equal to negative 1. That is true for any perpendicular lines. So find the slope of the line segment and solve for m1 x m2 = -1.
The line bisects the line segment, and that means it intersects the line segment at the midpoint, so use the midpoint formula to find that intersecting point, and now we have the slope and a point to work with,
First, the slope of the line segment, change of y divided by change of x, or rise over run. (-3 - 9 ) = -12 for the change of Y. (y2 -y1). For the run, (4 - 8) = -4 for change of x (x2 - x1). Change of y divided by change of x is -12/-4 = 3, giving the slope of the line segment. Now use that in the equation m1 x m2 = -1 to get the slope of the intersecting line.
3 x m2 = -1, divide both sides by 3, and
m2 = -1/3
We need a point on that line, and since the line intersects at the midpoint of the line segment, we can get that with the midpoint formula, (x1 + x2)/2 , (y1 + y2)/2. That gives us (8 + 4)/2 , (9 + (-3))/2, which is the point 6,6. Now we have a point and a slope and can use the point slope formula to get the line equation.
y - y1 = m(x - x1)
y - 6 = -1/3(x - 6), multiply both sides by -3 to get rid of the fractional coefficient,
-3y + 18 = x - 6, subtract 18 from both sides,
-3y = x - 24, divide both sides by -3 to get the equation in standard form
y = -1/3x + 8, final answer.
Hope that helps.
