The midpoint M of a segment [AB] is as far from A as it is from B. Is it the only point with that property, or are there others, away from the segment?
Every orange point above is the same distance from A as it is from B. Notice that they all lie on a straight line. Drawing that line through them gives the picture below.
The line passes through the midpoint M, and it is perpendicular to [AB]: it crosses the segment at a right angle.
The perpendicular bisector of a segment is the line that passes through the midpoint of the segment and is perpendicular to it.
Every point on the perpendicular bisector of [AB] is the same distance from A as from B, and no point off it is.
The name carries both conditions.
A line that satisfies one condition without the other is not the perpendicular bisector: a line through M at any other angle bisects the segment without being perpendicular to it, and a line at right angles to the segment anywhere other than M is perpendicular to it without bisecting it.
Checkpoint 1
The segment [AB] and four lines L1, L2, L3 and L4 are drawn on the grid below.
Which line is the perpendicular bisector of [AB]?
Checkpoint 2
Four points A, B, C and D and four lines L1, L2, L3 and L4 are drawn on the grid below.
Select every statement that is true.
A perpendicular bisector is a line, so it has an equation. The definition says how to find it: the line passes through the midpoint of the segment, and it is perpendicular to the segment. So find the gradient of the segment, use the fact that perpendicular lines have negative reciprocal gradients to get the gradient of the bisector, and then use the midpoint, a known point on the line, to write its equation.
Find the equation of the perpendicular bisector of [AB], where A(−1,−3) and B(5,1). Give the answer in the form y=mx+c.
The midpoint of [AB] is
The gradient of [AB] is
so the perpendicular bisector has gradient −23, the reciprocal of 32 with its sign changed. It passes through M(2,−1):
Checking: 32×(−23)=−1, so the line is perpendicular to the segment, and x=2 gives y=−3+2=−1, so it passes through the midpoint.
Checkpoint 3
Consider the points A(1,−2) and B(7,2).
Find the equation of the perpendicular bisector of [AB], giving your answer in the form y=mx+c.
When [AB] is horizontal, its gradient is 0 and there is no reciprocal to take: the perpendicular bisector is the vertical line through the midpoint, x=a where a is the midpoint's x-coordinate. When [AB] is vertical, its gradient is undefined, and the perpendicular bisector is the horizontal line through the midpoint, y=b where b is the midpoint's y-coordinate. For A(1,3) and B(7,3) the perpendicular bisector is x=4.
Checkpoint 4
Consider the points A(2,7), B(2,−1) and C(8,−1).
Find the equation of the perpendicular bisector of [AB] and the equation of the perpendicular bisector of [BC].
Exam-style problem 1
Two campsites in a nature reserve are marked A(2,2) and B(6,10) on the map below, where one unit represents one kilometre. A straight water pipeline runs across the reserve along the line x+y=13.
Find the equation of the perpendicular bisector of [AB]. Give your answer in the form y=mx+c.
A tap is to be installed on the pipeline at the point that is the same distance from both campsites.
Determine the coordinates of the point where the tap should be installed.
Find the distance, in kilometres, from the tap to campsite A.
Exam-style problem 2
Two villages are marked on a map, where one unit represents one kilometre. A straight cycle path is built so that every point on the path is the same distance from village A(3,2) as it is from village B. The path lies along the line y=−3x+21.
Find the equation of the line (AB). Give your answer in the form y=mx+c.
Determine the coordinates of the midpoint of [AB].
Hence find the coordinates of village B.
Nice work reaching the end of Perpendicular Bisectors. Here is what this lesson set out to teach - worth reading back with fresh eyes:
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