Symmetry & Transformations

Geometry is full of patterns and motion. A line of symmetry splits a shape into two halves that match exactly. A transformation moves a shape without changing its size or proportions. Together, symmetry and transformations describe how flat shapes look the same after they are folded, slid, flipped, or turned.

What Is a Line of Symmetry?

• A line of symmetry is an imaginary line that cuts a shape into two mirror-image halves.
• If you fold the shape along the line, the two halves match perfectly edge to edge.
• A shape can have zero, one, or many lines of symmetry.
• A square has 4 lines of symmetry; an equilateral triangle has 3; a circle has infinitely many.
• Shapes like the letter F or the letter R have 0 lines of symmetry — no fold makes the halves match.

The Three Basic Transformations

1. Translation (slide): every point of the shape moves the same distance in the same direction.
2. Reflection (flip): the shape is flipped across a line, like looking in a mirror.
3. Rotation (turn): the shape is rotated around a fixed point by some number of degrees.
4. In all three, the shape stays the same size and shape — it stays congruent to the original.

Everyday Words for the Three Moves

• Slide is the everyday word for translation.
• Flip is the everyday word for reflection.
• Turn is the everyday word for rotation.
• A turn can be a quarter-turn (90 degrees), a half-turn (180 degrees), or a three-quarter-turn (270 degrees).

How to Test for Symmetry

1. Pick a candidate line: vertical, horizontal, or diagonal through the shape.
2. Imagine folding the shape along that line.
3. If the two halves land exactly on top of each other, the line is a line of symmetry.
4. Check every possible line — some shapes have more than one.