An isosceles triangle folded along its axis of symmetry (dashed). One pair of mirror points P and P′ is joined by an amber segment — the axis cuts it exactly in half, at a right angle. The axis is the perpendicular bisector of every mirror-point pair.
Hold a paper shape up to a mirror, or fold it straight down the middle. If the two halves land exactly on each other, the shape has reflection symmetry, and the fold line is its axis of symmetry. Reflection is the third rigid motion — after the slide of Stage 14 and the turn — it flips a figure onto its mirror image without changing its size or shape one bit. Hiding inside every reflection is an old friend from our work with perpendicular lines: the perpendicular bisector. Meet it well here, because next lesson it unlocks the isosceles triangle and everything that follows from two equal sides.
15.4.1 Figures with reflection symmetry
Take a flat shape, fold it along a straight crease, and look: if the part on one side of the crease lands exactly on the part on the other side, the shape has reflection symmetry. That crease is called the axis of symmetry (some books say line of symmetry — same thing). The fold is the test: lay one half down on the other, and every point must find a partner.
How many axes a figure has depends on the figure. A plain scalene triangle has none — no fold lines up its three different sides. An isosceles triangle, or the capital letter A, has exactly one. A rectangle has two (one across, one up–down), a square has four, and a circle has infinitely many — every line through its center is a fold line.
From left: a scalene triangle (no axis), an isosceles triangle (1 axis), a rectangle (2 axes), and a square (4 axes). Each dashed line is a fold along which the figure matches itself.
Key idea
A line is an axis of symmetry of a figure exactly when folding along it makes the two halves coincide. A figure may have none, one, several, or infinitely many.
Watch out
Not every line through the "middle" is an axis. A diagonal of a rectangle (that isn't a square) splits it into two equal triangles — but fold along it and the corners do not match. An axis must be the fold that makes the halves land on top of each other, not merely cut the area in two.
15.4.2 Two figures that are reflections
Symmetry is one figure matching itself. The same flip can also carry one figure onto a different figure. Stand a triangle in front of a mirror line: across the line appears its mirror image. The flip that takes the first onto the second is a reflection across that line, and the line is again called the axis of the reflection.
A reflection is a rigid motion — it never stretches, shrinks, or bends. So a figure and its reflection are congruent (the very idea from 15.3): every side and every angle is preserved. There is just one twist — its orientation is flipped, the way your left hand is a mirror image of your right. Read the labels A B C around the original counterclockwise and you'll find A′ B′ C′ run clockwise.
△ABC and its reflection △A′B′C′ across the vertical axis. The two are congruent — same side lengths, same angles — but flipped: the lettering reverses its turning direction.
Mirror in the plane
On a coordinate grid the y-axis acts as a vertical mirror: the point (3, 2) reflects to (−3, 2) — the height stays, the left–right sign flips. In general, reflecting (a, b) across the y-axis gives (−a, b); across the x-axis it gives (a, −b). We'll use the grid version in 15.4.5.
15.4.3 The perpendicular-bisector property of the axis
Here is the single fact that makes reflection so useful. Pick any point P and let P′ be its mirror image across the axis. Join them with the segment PP′. Then the axis is the perpendicular bisector of PP′: it crosses the segment at its midpoint and at a right angle.
Why must this be true? Folding along the axis lays P exactly onto P′. So the crease passes squarely between them — equally far from each (the midpoint) and square to the line joining them (a right angle). That is precisely the definition of a perpendicular bisector. Said another way: every point sitting on the axis is the same distance from P as from P′.
Key idea
For a reflection, the axis is the perpendicular bisector of the segment joining each point to its image. Equivalently, every point on the axis is equidistant from a point and its mirror image.
Try it Drag P — watch its mirror image P′
Slide point P toward or away from the slate axis. Its image P′ always lands the same distance on the far side, and the axis slices PP′ in half at a right angle.
Move P sideways
P's height
15.4.4 The perpendicular bisector of a segment
Section 15.4.3 says points on the axis are equidistant from P and P′. Flip that statement around and it becomes a clean, distance-only definition:
Definition
The perpendicular bisector of a segment AB is the set of all points that are equidistant from A and B. A point lies on it if and only if its distance to A equals its distance to B.
This is the same line you build with compass and straightedge in 15.2: swing equal arcs from A and from B, and the two crossing points are each equidistant from A and B — join them and you have the perpendicular bisector. It runs straight through the midpoint of AB, square to it. Later, this set-of-equal-distances idea is exactly how you find the center of a circle through three points.
Try it Equal distances land on the bisector
Drag the test point M. When its distance to A equals its distance to B, it is sitting exactly on the perpendicular bisector — both arms light green.
Move M across
M's height
15.4.5 Drawing symmetric figures on a grid
Graph paper turns reflection into simple counting. To reflect a figure across a vertical axis line, take it one vertex at a time: count how many squares a corner sits from the axis, then plant its image the same number of squares on the other side, at the same height. Do that for every corner, then reconnect them in the same order. A vertical axis gives a left↔right flip; a horizontal axis gives an up↔down flip.
Because reflection is a rigid motion, the image you build is congruent to the original — same side lengths, same angles — only its handedness is reversed, exactly as in 15.4.2. This is the third rigid motion alongside the slide of Stage 14 and the turn; together they are the moves that decide whether two figures are congruent.
Try it Reflect a polygon across a grid line
Choose a vertical or horizontal axis. Each corner of the green image is the same number of squares across the axis as its blue partner — amber connectors show the matched distances.
Axis
Watch out
A reflection is not a slide and not a turn. A slide keeps the figure facing the same way; a reflection flips its handedness. If your "reflected" triangle reads the same way around as the original, you slid it — you didn't fold it.
★ Recap
What to carry forward
• An axis of symmetry is a fold line along which a figure's two halves coincide; a figure can have none, one, several, or infinitely many.
• A reflection is the third rigid motion. A figure and its mirror image are congruent, just with flipped orientation (left hand ↔ right hand).
• The axis is the perpendicular bisector of the segment joining each point to its image — it cuts PP′ at its midpoint, at a right angle.
• The perpendicular bisector of AB is the set of points equidistant from A and B.
• On a grid, reflect each vertex the same number of squares across the axis: (a, b) → (−a, b) across the y-axis, (a, −b) across the x-axis.
✎ Exercises
How many axes of symmetry does each shape have: a square, a rectangle (not a square), and a scalene triangle?
Answer
A square has 4 (two through opposite sides, two through opposite corners). A non-square rectangle has 2 (the two midlines only — the diagonals are not axes). A scalene triangle has 0.
Reflect the point (3, 2) across the y-axis. What are the image's coordinates?
Answer
The y-axis is a vertical mirror, so the left–right sign flips and the height stays: (−3, 2). In general, (a, b) → (−a, b) across the y-axis.
Fill in the blank: the axis of symmetry is the ______ of the segment joining a point to its image.
Answer
The perpendicular bisector. It crosses the segment PP′ at its midpoint and at a right angle.
A point M lies on the perpendicular bisector of segment AB, and you measure MA = 6.5 cm. What is MB, and why?
Answer
MB = 6.5 cm. Every point on the perpendicular bisector of AB is equidistant from A and B — that is the defining property — so MA = MB.
Reflect the triangle with vertices (1, 1), (4, 1), (1, 3) across the y-axis. List the image vertices.
Answer
Flip each x-coordinate's sign and keep the height: (−1, 1), (−4, 1), (−1, 3). The image is congruent to the original, just flipped left–right.
Explain why a figure and its reflection are always congruent, yet you can tell them apart.
Answer
A reflection is a rigid motion: it preserves every length and angle, so original and image have the same size and shape — congruent. But it reverses orientation (handedness), so the labels run the opposite way around — like a left hand versus a right hand. They match in measure but not in turning direction.
🎯 Quick check
Six questions to lock it in. Tap the answer you think is right.
§ For teachers and parents
Reflection is the rigid motion that finally explains the isosceles triangle, so this lesson is the hinge between congruence (15.3) and the isosceles family (15.5). The big idea to make concrete is that the axis of symmetry is the perpendicular bisector of every mirror-point pair — and that "perpendicular bisector" has a pure distance meaning: it is exactly the set of points equidistant from the two endpoints. The two interactive figures (mirror and equidistant point) are built to show those two statements as the same fact from two directions.
Watch for two common misconceptions. First, students often confuse a reflection with a slide or a turn — the giveaway is orientation: a true reflection flips handedness, so the labels reverse their turning direction. Use the grid widget and ask "which way do A, B, C run now?" Second, many believe any line through the middle is an axis of symmetry; the diagonal of a non-square rectangle is the classic trap — it halves the area but the corners do not land on each other. Insist on the fold test, and on the perpendicular-bisector-of-matching-points definition.
This lesson supports Common Core 4.G.A.3 (recognize a line of symmetry), 8.G.A.1 and 8.G.A.3 (reflections as rigid motions and the coordinate rules for them), and HS G-CO.A.3, G-CO.A.4, and G-CO.A.5 (define reflections precisely, describe the lines of symmetry that carry a figure onto itself, and specify a reflection that maps one figure to another).
eastmath.com · Stage 15 · 15.4 Reflection Symmetry · Reasoning, one step at a time