Question

NO LINKS!! URGENT HELP PLEASE!!!

Find a formula that expresses the fact that an arbitrary point P(x, y) is on the perpendicular bisector "l" of segment AB.

A(-6, 4), B(14, -12)

Answer 1

The perpendicular bisector of a line segment AB is the line that passes through the midpoint of AB and is perpendicular to AB.

To find the equation of the perpendicular bisector of segment AB, we can follow these steps:

• Find the midpoint M of AB. The coordinates of M are:

M = ( (x1 + x2)/2, (y1 + y2)/2 )

where A = (x1, y1) and B = (x2, y2).

• In this case, A = (-6, 4) and B = (14, -12), so the coordinates of M are:

M = ( (-6 + 14)/2, (4 - 12)/2 ) = (4, -4)

• Find the slope m of AB. The slope of AB is:

m = (y2 - y1) / (x2 - x1)

• In this case, the slope of AB is:

m = (-12 - 4) / (14 - (-6)) = -16/20 = -4/5

• Find the slope of the line that is perpendicular to AB. The slope of a line perpendicular to AB is the negative reciprocal of the slope of AB. So, the slope of the line that is perpendicular to AB is:

m_perp = -1/m

• In this case, the slope of the line that is perpendicular to AB is:

m_perp = -1/(-4/5) = 5/4

• Use the point-slope form of the equation of a line to find the equation of the perpendicular bisector. The point-slope form of the equation of a line is:

y - y1 = m(x - x1)

• We can use the midpoint M as the point (x1, y1) and the slope m_perp as the slope m:

y - (-4) = (5/4)(x - 4)

Simplifying, we get:

y + 4 = (5/4)x - 5

• Moving terms around, we get:

(5/4)x - y - 9 = 0

So, the formula that expresses the fact that an arbitrary point P(x, y) is on the perpendicular bisector of segment AB is:

(5/4)x - y - 9 = 0