Reflections and Rotations, Year 8 Maths

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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 isometriestransformations 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

ReflectionA flip of a shape over a mirror line, every point maps to an equal distance on the other side

RotationA turn of a shape about a fixed centre of rotation by a given angle and direction

Axis of symmetryThe mirror line used in a reflection

Centre of rotationThe fixed point about which a shape is rotated

IsometryA transformation that preserves distances and angles (size and shape stay the same)

Invariant pointA point that does not move under a transformation (e.g. the origin under any rotation about the origin)

⚠ 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?

Reflect over x-axis: $(x,y)\to(x,-y)$

Reflect over y-axis: $(x,y)\to(-x,y)$

Reflect over $y=x$: $(x,y)\to(y,x)$

Rotate 90° ACW: $(x,y)\to(-y,x)$

Rotate 180°: $(x,y)\to(-x,-y)$

Rotate 270° ACW (= 90° CW): $(x,y)\to(y,-x)$

Badges This Lesson

Mirror Master

Reflection Ruler

Rotation Hero

Origin Orbiter

Transform Titan

Symmetry Seeker

Mark lesson complete

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