Key takeaway: A perpendicular bisector of a segment is the line through its midpoint at a right angle to it, and each point of that line lies equally far from the segment’s two ends.
Grade 10 – GeometryG-CO.C.9Geometry and Measurement
Definition of Perpendicular Bisector
A perpendicular bisector works as a divider and a square crossing in one: the line passes through the midpoint of a segment and stands at \(90^{\circ}\) to it, leaving two halves of equal length.
Fold a paper strip so one end lands on the other; the crease is the line. Points on the crease are equally far from both ends, and points off the crease are closer to one end. Around a triangle, the three creases from its sides meet at the circumcenter.
Formula and notation
For endpoints \(A\) and \(B\), a point \(P\) lies on the perpendicular bisector exactly when:
\[ PA = PB \]
On a coordinate plane, use the midpoint of the segment together with the negative reciprocal of its slope \(m\), which is \(-\frac{1}{m}\).
Key facts
- The line passes through the midpoint of the segment and forms four right angles there.
- Points on it are equidistant from the endpoints, and no other points are.
- The slopes of a segment and of this line multiply to negative 1.
- Three of them, one per side of a triangle, share a single crossing point.
Perpendicular Bisector example
A sprinkler must be equally far from two fence posts at \(A(-4, 3)\) and \(B(2, -1)\), measured in yards on a grid. Write the equation of the line of all such positions.
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Midpoint: \((\frac{-4+2}{2},\ \frac{3+(-1)}{2}) = (-1, 1)\).
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Rise over run between the posts: \(\frac{-1-3}{2-(-4)} = -\frac{4}{6}\). Turn it over and change the sign: \(\frac{6}{4} = 1.5\) is the sprinkler slope.
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Run the line through the midpoint: \(y – 1 = 1.5(x + 1)\), which gives \(y = 1.5x + 2.5\).
The sprinkler can go anywhere on \(y = 1.5x + 2.5\). For \(P(1, 4)\): \(PA^2 = 25 + 1 = 26\) and \(PB^2 = 1 + 25 = 26\).
Common mistake
Watch out
Jonas sketches a line of slope \(\frac{2}{3}\) through \((-1, 1)\) and labels it the bisector of \(\overline{AB}\), reasoning that the opposite sign gives a perpendicular.
Why it is wrong
Changing the sign alone leaves the numerator and denominator in place, so the angle is wrong. The fence has slope \(-\frac{2}{3}\); a right-angle partner needs the fraction turned upside down and the sign changed, giving \(\frac{3}{2}\).
Related terms and lessons
- Geometry · Geometric ConstructionsLessonWorksheetQuiz
- Geometry · Triangle Theorems and Points of ConcurrencyLessonWorksheetQuiz
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