Perpendicular bisector
A perpendicular bisector is a line that both bisects a segment (cuts it into two equal halves) and is perpendicular (meets at 90°) to it in Grade 6 math (Saxon Math, Course 1). Any point on the perpendicular bisector is equidistant from both endpoints of the segment. To construct one with a compass: draw arcs from each endpoint with the same radius (greater than half the segment), then connect the two intersection points. This line is both the axis of symmetry of the segment and the locus of all points equidistant from the two endpoints. The perpendicular bisector concept is foundational for circumscribing circles about triangles.
Key Concepts
A perpendicular bisector is a line that bisects a segment and is also perpendicular to it.
Step 1: To find the perpendicular bisector of segment $XY$, set your compass to a width more than half of $XY$'s length. Step 2: Swing arcs from point $X$ above and below the segment, then repeat from point $Y$ without changing the compass width. Step 3: Draw a straight line connecting the two points where the arcs intersect to form the perpendicular bisector.
Imagine a line segment is a tightrope. A perpendicular bisector is like a support pole that is placed exactly in the middle of the rope and stands perfectly straight up at a 90 degree angle. It's the ultimate balance point, providing perfect support by cutting the segment into two equal halves at a perfect right angle.
Common Questions
What is a perpendicular bisector?
A line that crosses a segment at its exact midpoint AND forms a 90° angle with the segment.
What is special about points on a perpendicular bisector?
Every point on the perpendicular bisector is exactly the same distance from both endpoints of the original segment.
How do you construct a perpendicular bisector with a compass?
Set compass to more than half the segment length. Draw arcs from each endpoint. Connect the two arc intersection points. That line is the perpendicular bisector.
What two conditions must a line meet to be a perpendicular bisector?
It must pass through the midpoint of the segment (bisect it) AND be perpendicular (90°) to the segment.
How is the perpendicular bisector used in circumscribed circles?
The circumcenter of a triangle (center of the circumscribed circle) is the intersection of the perpendicular bisectors of all three sides.