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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?
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.
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.
When reflecting a point $(x, y)$ over each standard axis:
Memory tip: reflect over the x-axis → x stays, y flips. Reflect over the y-axis → y stays, x flips.
Triangle with vertices $A(1,2)$, $B(3,2)$, $C(2,4)$ is reflected over the x-axis.
Rule: $(x,\; y) \to (x,\; -y)$
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.
Rotating the point $(x, y)$ about the origin $(0,0)$:
Trick: For 90° ACW, swap the coordinates and negate the new x-coordinate: $(x,y)\to(-y,x)$.
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.
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 $A(1,3)$, $B(4,3)$, $C(4,1)$.
Rule: $(x,y)\to(-x,y)$
$P(2,1)\to P'(-2,1)$
$Q(5,1)\to Q'(-5,1)$
$R(4,4)\to R'(-4,4)$
270° ACW (= 90° CW): $(x,y)\to(y,-x)$
$T(5,-2)\to T'(-2,-5)$
90° CW = 270° ACW: $(x,y)\to(y,-x)$
$A(1,3)\to A'(3,-1)$
$B(4,3)\to B'(3,-4)$
$C(4,1)\to C'(1,-4)$
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?