Reflections and Rotations
Apply the coordinate rules for reflecting in the axes and rotating about the origin, work through both on a triangle, and identify the points that do not move.
Printable Worksheets
Print or save as PDF, or build a custom worksheet from any module's questions.
Think First
Think of a mirror image, if you stand 2 metres from a mirror, where does your image appear? What happens to left and right? What about up and down?
Reflections and Rotations
Both reflections and rotations are isometries transformations that preserve the size and shape of a figure. Reflections flip a shape over a mirror line; rotations turn a shape about a fixed point called the centre of rotation. In both cases the image is congruent to the object.
What You'll Master
- Apply reflection rules over the x-axis, y-axis, and the line $y = x$
- Apply rotation rules (90°, 180°, 270° anticlockwise) about the origin
- Find the image coordinates of points and shapes after reflections and rotations
- Identify which transformation maps an object to its image
Words You Need
⚠ Spot the Trap
For rotations, anticlockwise is the positive direction in mathematics. "90° anticlockwise" and "270° clockwise" are the same rotation. Always state the direction clearly. Also: after reflection over the y-axis, the x-coordinate changes sign but y stays the same, students often flip both coordinates by mistake.
Reflection Coordinate Rules
When reflecting a point $(x, y)$ over each standard axis:
- Over the x-axis: $(x,\; y) \to (x,\; -y)$, y-coordinate changes sign, x stays
- Over the y-axis: $(x,\; y) \to (-x,\; y)$, x-coordinate changes sign, y stays
- Over $y = x$: $(x,\; y) \to (y,\; x)$, x and y coordinates swap
Memory tip: reflect over the x-axis → x stays, y flips. Reflect over the y-axis → y stays, x flips.
Worked Example, Reflecting a Triangle
Triangle with vertices $A(1,2)$, $B(3,2)$, $C(2,4)$ is reflected over the x-axis.
Rule: $(x,\; y) \to (x,\; -y)$
- $A(1,2) \to A'(1,\; -2)$
- $B(3,2) \to B'(3,\; -2)$
- $C(2,4) \to C'(2,\; -4)$
The image is the same shape, mirrored below the x-axis. Each point is the same distance from the x-axis as the original, but on the opposite side.
Rotation Rules About the Origin
Rotating the point $(x, y)$ about the origin $(0,0)$:
- 90° anticlockwise: $(x,\; y) \to (-y,\; x)$
- 180° (either direction): $(x,\; y) \to (-x,\; -y)$
- 270° anticlockwise (= 90° clockwise): $(x,\; y) \to (y,\; -x)$
- 360°: $(x,\; y) \to (x,\; y)$, back to the start
Trick: For 90° ACW, swap the coordinates and negate the new x-coordinate: $(x,y)\to(-y,x)$.
Worked Example, Rotating a Point
Rotate point $P(3, 1)$ about the origin.
90° anticlockwise: $(3,1) \to (-1,\; 3)$
180°: $(3,1) \to (-3,\; -1)$
270° anticlockwise (= 90° clockwise): $(3,1) \to (1,\; -3)$
Notice each successive 90° rotation moves the point to the next quadrant, tracing a circle around the origin.
Common Pitfalls
- Reflecting over x-axis: forgetting that only the y-coordinate changes sign
- Reflecting over y-axis: changing both coordinates instead of just x
- Rotating 90° ACW: using $(y, -x)$ instead of $(-y, x)$, swap first, then negate
- Confusing clockwise and anticlockwise directions
- Forgetting that after reflection the orientation (handedness) of the shape is reversed
Copy This Into Your Book
Reflections: x-axis: $(x,y)\to(x,-y)$ | y-axis: $(x,y)\to(-x,y)$ | $y=x$: $(x,y)\to(y,x)$
Rotations about origin: 90° ACW: $(x,y)\to(-y,x)$ | 180°: $(x,y)\to(-x,-y)$ | 270° ACW: $(x,y)\to(y,-x)$
Both are isometries: size and shape are always preserved.
Point $(4, -3)$ is reflected over the x-axis. What are the image coordinates?
Point $(-2, 5)$ is reflected over the y-axis. What are the image coordinates?
Point $(2, 3)$ is rotated 90° anticlockwise about the origin. What is the image?
Point $(-1, 4)$ is rotated 180° about the origin. What is the image?
Triangle $A(1,2)$, $B(3,2)$, $C(2,4)$ maps to $A'(1,-2)$, $B'(3,-2)$, $C'(2,-4)$. Which transformation was applied?
Q6. Triangle $PQR$ has vertices $P(2, 1)$, $Q(5, 1)$, $R(4, 4)$. Reflect the triangle over the y-axis. List the image vertices $P'$, $Q'$, $R'$.
Q7. Point $T(5, -2)$ is rotated 270° anticlockwise about the origin. State the coordinate rule for this rotation and find the image $T'$.
Q8. A shape is rotated 90° clockwise about the origin. Write the coordinate rule for this rotation (hint: 90° CW = 270° ACW), then apply it to vertices $A(1,3)$, $B(4,3)$, $C(4,1)$.
Show Answers
Q6
Rule: $(x,y)\to(-x,y)$
$P(2,1)\to P'(-2,1)$
$Q(5,1)\to Q'(-5,1)$
$R(4,4)\to R'(-4,4)$
Q7
270° ACW (= 90° CW): $(x,y)\to(y,-x)$
$T(5,-2)\to T'(-2,-5)$
Q8
90° CW = 270° ACW: $(x,y)\to(y,-x)$
$A(1,3)\to A'(3,-1)$
$B(4,3)\to B'(3,-4)$
$C(4,1)\to C'(1,-4)$
Stretch Challenge
A shape is first reflected over the x-axis, then reflected over the y-axis. Using a specific example, say triangle with vertex $A(2, 3)$, show that the combined result is identical to a single 180° rotation about the origin. Does the order of the two reflections matter?