Lesson 1 — Line Symmetry

Big idea: A figure has line symmetry when a line divides it into two mirror-image halves.

Words to know

Word	Meaning
Perpendicular bisector	A line perpendicular to a segment at its midpoint.
Equidistant	At the same distance from two points.
Symmetric points	Two points are symmetric with respect to a line if the line is the perpendicular bisector of the segment joining them.
Line symmetry	A figure can be folded along a line so that the two halves coincide.

Every point on the perpendicular bisector of AB is equidistant from A and B.
Theorem 16: In a plane, two points each equidistant from the endpoints of a segment determine the perpendicular bisector of the segment.
A figure has line symmetry if a reflection over a line maps the figure onto itself.
The fold line is the line of symmetry.
Construction 6: Construct a line perpendicular to a given line through a given point.

If the homework asks…	Do this…
Find the perpendicular bisector	Find the midpoint; draw a perpendicular line.
Decide whether points are symmetric	Check whether the line is the perpendicular bisector of the segment.
Use equidistance	Any point on the perpendicular bisector is equally far from both endpoints.
Find lines of symmetry	Fold or reflect; check whether the halves match.
Construct a perpendicular	Use Construction 6.
Prove a line is the perpendicular bisector	Show it is perpendicular and passes through the midpoint.

A and B are endpoints of a segment. Points P and Q are both equidistant from A and B.

By Theorem 16, the line through P and Q is the perpendicular bisector of AB. So it crosses AB at its midpoint and forms right angles.

A perpendicular bisector must pass through the midpoint, not just any point.
“Equidistant” means equal distances to the two endpoints.
A line of symmetry is not always horizontal or vertical.
A figure can have more than one line of symmetry.