Reflections: Step-by-Step Guide
A reflection flips a shape over a line, like a mirror. The line is called the line of reflection. Each point and its mirror image are the same distance from the line, on opposite sides. The most common reflections: over the x-axis, (x, y) becomes (x, -y). Over the y-axis, (x, y) becomes (-x, y). Over the line y = x, (x, y) becomes (y, x).
Math Racer skill guide · Geometry · Grade 10
How to solve it, step by step
- Find the line of reflection. Decide which line you are reflecting over: the x-axis, the y-axis, y = x, y = -x, or another horizontal or vertical line.\[\text{line of reflection: } y = 0,\; x = 0,\; y = x,\; \dots\]
- Use the reflection rule. Over the x-axis: (x, y) becomes (x, -y). Over the y-axis: (x, y) becomes (-x, y). Over y = x: (x, y) becomes (y, x). Over y = -x: (x, y) becomes (-y, -x).\[\begin{aligned}(x, y) &\mapsto (x, -y) \;\text{over x-axis}\\(x, y) &\mapsto (-x, y) \;\text{over y-axis}\\(x, y) &\mapsto (y, x) \;\text{over } y = x\end{aligned}\]
- Find the new coordinates. Use the rule on each corner (vertex) of the shape.\[A(x, y) \mapsto A'(\dots)\]
- Check the distances. Each point and its image should be the same distance from the line of reflection, and the line connecting them should make a right angle with the mirror line.\[d(A, \ell) = d(A', \ell)\]
Worked example
Triangle ABC has vertices A(2, 3), B(5, 1), C(4, 6). Reflect the triangle over the x-axis and find the image vertices.
Step 1: The line of reflection is the x-axis.
Step 2: The rule is (x, y) becomes (x, -y). The y-coordinate changes sign.
Step 3: A(2, 3) becomes A'(2, -3).
Step 4: B(5, 1) becomes B'(5, -1).
Step 5: C(4, 6) becomes C'(4, -6).
Check: Each point went from above the x-axis to the same distance below it. The shape has the same size and angles.
The new vertices are A'(2, -3), B'(5, -1), and C'(4, -6).
Tips and common mistakes
- A reflection keeps the same size and shape. The image is congruent to the original.
- A reflection flips the direction of the shape. If you go around the corners clockwise in the original, you go around counterclockwise in the image.
- Over the x-axis, change the sign of y. Over the y-axis, change the sign of x. These are easy to mix up, so write the rule down first.
- Over y = x, swap x and y. Over y = -x, swap them and change both signs.
- Over a horizontal line y = b: (x, y) becomes (x, 2b - y). Over a vertical line x = a: (x, y) becomes (2a - x, y).
- A point that is on the line of reflection does not move.
- Name the image points with a prime mark (A', B', C') and sketch the picture to check that it looks like a mirror image.