Reflections

A reflection flips a figure over a line to make a mirror image. The line is called the line of reflection. Each point and its image are the same distance from the line, on opposite sides. Reflecting over the x-axis uses the rule (x, y) \to (x, -y), and over the y-axis (x, y) \to (-x, y).

Before this lesson: You should know Translations, the Coordinate Plane and Horizontal and Vertical Lines.

What Is a Reflection?

Stand in front of a mirror and raise your right hand. Your mirror image raises the hand on the other side. It looks the same size, but it is flipped. A reflection in geometry works the same way.

Think of folding a sheet of paper along a line. A point and its image would land on top of each other. That fold line is the line of reflection.

A reflection is a rigid motion, like a translation. Lengths and angles stay the same, so the image is congruent to the pre-image. One thing does change: the figure is flipped. If you read the vertices A, B, C in clockwise order, the image goes counterclockwise.

The segment from a point to its image always crosses the line of reflection at a right angle. The line cuts that segment exactly in half. So the line of reflection is the perpendicular bisector of the segment.

Reflection Rules

On a coordinate grid, these rules give the image of any point (x, y).

Line of reflection	Rule	Example
x-axis	(x, y) \to (x, -y)	(2, 5) \to (2, -5)
y-axis	(x, y) \to (-x, y)	(2, 5) \to (-2, 5)
The line y = x	(x, y) \to (y, x)	(2, 5) \to (5, 2)
The line y = -x	(x, y) \to (-y, -x)	(2, 5) \to (-5, -2)
Vertical line x = h	(x, y) \to (2h - x, y)	Over x = 1: (2, 5) \to (0, 5)
Horizontal line y = k	(x, y) \to (x, 2k - y)	Over y = 1: (2, 5) \to (2, -3)

A memory trick: reflecting over the x-axis changes the y-coordinate. The x-axis is the mirror, so x stays put. The same idea works for the y-axis.

How to Reflect a Figure

1. Find the line of reflection.
2. For each vertex, pick the matching rule from the table. For other lines, count the distance from the point straight to the line.
3. Find the image of every vertex.
4. Plot the image points and connect them in the same order.
5. Label the image with prime marks.
6. Check: each point and its image should be the same distance from the line, on opposite sides.

Worked Examples

Example 1: Reflect a point over each axis

Reflect P(3, -5) over the x-axis. Then reflect P over the y-axis.

1. Over the x-axis, use (x, y) \to (x, -y). Keep x: 3
2. Take the opposite of y: -(-5) = 5
3. Image over the x-axis: (3, 5)
4. Over the y-axis, use (x, y) \to (-x, y). Take the opposite of x: -(3) = -3
5. Keep y: −5
6. Image over the y-axis: (-3, -5)

Answer: Over the x-axis, (3, 5). Over the y-axis, (-3, -5).

Example 2: Reflect a triangle over the y-axis

Triangle ABC has vertices A(1, 1), B(4, 1) and C(2, 4). Reflect it over the y-axis. This is the triangle in the graph above.

Vertex	Take the opposite of x	Keep y	Image
A(1, 1)	-(1) = -1	1	A'(-1, 1)
B(4, 1)	-(4) = -4	1	B'(-4, 1)
C(2, 4)	-(2) = -2	4	C'(-2, 4)

Check C. Point C is 2 units right of the y-axis. Point C' is 2 units left of it. They match. Also notice the order. Going A to B to C turns one way. Going A′ to B′ to C′ turns the other way, because the triangle flipped.

Answer: A'(-1, 1), B'(-4, 1), C'(-2, 4)

Example 3: Reflect over the line y = x

Triangle JKL has vertices J(-2, 5), K(3, 1) and L(0, -4). Reflect it over the line y = x.

The rule is (x, y) \to (y, x). Swap the two coordinates.

1. Swap the coordinates of J: J(-2, 5) \to J'(5, -2)
2. Swap the coordinates of K: K(3, 1) \to K'(1, 3)
3. Swap the coordinates of L: L(0, -4) \to L'(-4, 0)
4. Check J. Find the midpoint of JJ' (see midpoint formula): \left(\frac{-2 + 5}{2}, \frac{5 + (-2)}{2}\right)
5. Add the numbers on top: \left(\frac{3}{2}, \frac{3}{2}\right)
6. The two coordinates are equal, so the midpoint is on the line y = x. The check works.

Answer: J'(5, -2), K'(1, 3), L'(-4, 0)

Example 4: Reflect over a vertical line

Reflect M(5, 3) and N(-1, -2) over the line x = 2.

The line is vertical, so only the x-coordinate changes. Count across to the line, then the same distance past it.

1. Distance from M to the line: 5 - 2 = 3. M is 3 units right of the line.
2. Go 3 units left of the line: 2 - 3 = -1
3. Keep y: 3. So M'(-1, 3).
4. Distance from N to the line: 2 - (-1) = 2 + 1 = 3. N is 3 units left of the line.
5. Go 3 units right of the line: 2 + 3 = 5
6. Keep y: −2. So N'(5, -2).

You can also use the rule (x, y) \to (2h - x, y) with h = 2. For M: 2(2) - 5 = 4 - 5 = -1. It gives the same answer.

Answer: M'(-1, 3) and N'(5, -2)

Example 5: Word problem (shortest path)

A camper’s tent is at A(1, 4). A river runs along the x-axis. She walks from the tent to the river to fill a bucket. Then she walks to the campfire at B(7, 2). Where should she touch the river to make her walk as short as possible?

Here is the trick. Reflect B over the river. Any path to the river and then to B is just as long as the path to the river and then to B'. The shortest path to B' is a straight line. So find where segment AB' crosses the river.

1. Reflect B over the x-axis: B(7, 2) \to B'(7, -2)
2. Find the slope of AB' (see slope formula): m = \frac{-2 - 4}{7 - 1}
3. Subtract: m = \frac{-6}{6}
4. Divide: m = -1
5. Write the line through A(1, 4) (see point-slope form): y - 4 = -1(x - 1)
6. The river is where y = 0. Substitute 0 for y: 0 - 4 = -1(x - 1)
7. Subtract on the left: -4 = -1(x - 1)
8. Divide both sides by −1: 4 = x - 1
9. Add 1 to both sides: 5 = x

She should touch the river at (5, 0). How long is the walk? It equals the length of AB'.

1. Use the distance formula (see distance formula): AB' = \sqrt{(7 - 1)^2 + (-2 - 4)^2}
2. Subtract inside the parentheses: AB' = \sqrt{6^2 + (-6)^2}
3. Square: AB' = \sqrt{36 + 36}
4. Add: AB' = \sqrt{72}
5. Simplify (see simplifying square roots): \sqrt{72} = \sqrt{36 \cdot 2} = 6\sqrt{2}
6. Estimate with \sqrt{2} \approx 1.41: 6 \times 1.41 = 8.46

Answer: She should touch the river at (5, 0). The shortest walk is 6\sqrt{2} \approx 8.46 units.

Common Mistakes

Mistake	Example of the error	Fix
Changing the wrong coordinate	Over the x-axis, (3, -5) \to (-3, -5)	Over the x-axis, x stays and y flips: (3, 5)
Changing both signs over an axis	Over the y-axis, (2, 5) \to (-2, -5)	Only x changes: (-2, 5)
Mixing up y = x and y = -x	Over y = x, (2, 5) \to (-5, -2)	Over y = x, just swap: (5, 2)
Taking the opposite over x = h	Over x = 2, (5, 3) \to (-5, 3)	Count to the line and past it: (-1, 3)
Measuring at a slant	Counting a diagonal distance to the line	Measure straight across, at a right angle to the line.
Forgetting the flip	Labeling the image in the wrong order	Match each point to its own image: A to A′, B to B′.

Finding the Line of Reflection

Sometimes you know a point and its image, and you need the line. The line of reflection is the perpendicular bisector of the segment between them. It goes through the midpoint.

• If the point and image have the same y-coordinate, the line is vertical. Average the x-coordinates. For (-4, 3) and (2, 3), the line is x = \frac{-4 + 2}{2} = -1.
• If they have the same x-coordinate, the line is horizontal. Average the y-coordinates.
• If the coordinates are swapped, like (6, -2) and (-2, 6), the line is y = x.

Reflections also explain why some figures look the same on both sides. A figure with a line of symmetry matches its own reflection. You will see this in the Symmetry lesson later in this unit.

Practice Problems

1. Reflect (4, -7) over the x-axis.
2. Reflect (-6, 2) over the y-axis.
3. Reflect (-3, 8) over the line y = x.
4. Reflect (5, -1) over the line y = -x.
5. Triangle DEF has vertices D(2, 1), E(5, 3) and F(2, 6). Reflect it over the x-axis.
6. Reflect (7, 4) over the line x = 3.
7. Reflect (-2, 5) over the line y = 1.
8. Point A(-4, 3) reflects to A'(2, 3). Find the line of reflection.
9. The point (6, -2) maps to (-2, 6). Which line of reflection does this?
10. A pool ball sits at (1, 2). It bounces off the bottom cushion, the x-axis, and rolls into a pocket at (9, 6). Where does it hit the cushion? How far does it roll?

Show answers

1. Keep x: 4
   Opposite of y: -(-7) = 7
   Image: (4, 7)
2. Opposite of x: -(-6) = 6
   Keep y: 2
   Image: (6, 2)
3. Swap the coordinates: (8, -3)
4. Use (x, y) \to (-y, -x)
   New x is the opposite of y: -(-1) = 1
   New y is the opposite of x: -(5) = -5
   Image: (1, -5)
5. Keep each x and take the opposite of each y
   D(2, 1) \to D'(2, -1)
   E(5, 3) \to E'(5, -3)
   F(2, 6) \to F'(2, -6)
6. Distance to the line: 7 - 3 = 4 (4 right)
   Go 4 left of the line: 3 - 4 = -1
   Keep y: (-1, 4)
7. Distance to the line: 5 - 1 = 4 (4 above)
   Go 4 below the line: 1 - 4 = -3
   Keep x: (-2, -3)
8. Same y-coordinate, so the line is vertical
   Add the x-coordinates: -4 + 2 = -2
   Divide by 2: -2 \div 2 = -1
   Line: x = -1
9. The coordinates are swapped, so the line is y = x
10. Reflect the pocket over the x-axis: (9, -6)
    Slope from (1, 2): m = \frac{-6 - 2}{9 - 1} = \frac{-8}{8} = -1
    Line: y - 2 = -1(x - 1)
    Set y = 0: 0 - 2 = -1(x - 1)
    Subtract: -2 = -1(x - 1)
    Divide by −1: 2 = x - 1
    Add 1: x = 3, so it hits at (3, 0)
    Distance: \sqrt{(9 - 1)^2 + (-6 - 2)^2} = \sqrt{8^2 + (-8)^2}
    Square and add: \sqrt{64 + 64} = \sqrt{128}
    Simplify: \sqrt{64 \cdot 2} = 8\sqrt{2} units

FAQ

What is a reflection in geometry?

A reflection flips a figure over a line to make a mirror image. Each point and its image are the same distance from the line.

What is the rule for a reflection over the x-axis?

The rule is (x, y) \to (x, -y). Keep x and take the opposite of y.

What is the rule for a reflection over y = x?

The rule is (x, y) \to (y, x). Swap the x- and y-coordinates.

Is a reflection a rigid motion?

Yes. It keeps all lengths and angle measures, so the image is congruent. It does reverse the order of the vertices.

Summary

• A reflection flips a figure over a line of reflection.
• Each point and its image are the same distance from the line, on opposite sides.
• Over the x-axis: (x, -y). Over the y-axis: (-x, y).
• Over y = x: (y, x). Over y = -x: (-y, -x).
• For other lines, count straight to the line, then the same distance past it.
• A reflection is a rigid motion, but it flips the order of the vertices.

Previous lesson: Translations

Next lesson: Rotations

Back to the Geometry Index