G7 - Reflections, rotations and translations

You already know

Coordinates give horizontal position first, then vertical position. A vertical line has equation $x=\text{constant}$; a horizontal line has equation $y=\text{constant}$. Congruent shapes have equal corresponding lengths and angles.

Reflection

A transformation that sends every point to the opposite side of a mirror line at the same perpendicular distance. The image of $A$ is labelled $A'$, read “A prime”. Points on the mirror line do not move.

Reflections in axis-parallel lines

For a vertical mirror $x=h$,

For a horizontal mirror $y=k$,

Each mirror coordinate is the midpoint of a point's coordinate and its image's coordinate. In particular, reflection in the $y$-axis changes the sign of $x$, and reflection in the $x$-axis changes the sign of $y$.

Not on the formulae sheet: learn it.

Worked example 1: a vertical mirror away from the axis

Question. Triangle $ABC$ has vertices $A=(1,1)$, $B=(4,1)$, $C=(2,3)$. Reflect it in $x=-1$. Write down the image coordinates. Non-calculator.

Plan. Keep each height and put each vertex equally far to the left of the vertical mirror.

Why this step? The mirror is $x=-1$, so changing just the sign of $x$ would reflect in the wrong line.

Answer. $A'=(-3,1)$, $B'=(-6,1)$, $C'=(-4,3)$.

Check. The midpoint of each point and image has $x=-1$, and corresponding heights agree.

Worked example 2: a horizontal mirror

Question. Reflect the same triangle $A=(1,1)$, $B=(4,1)$, $C=(2,3)$ in $y=-1$. Write down the image coordinates. Non-calculator.

Plan. Keep each horizontal coordinate and put each image equally far below the mirror.

Why this step? A horizontal mirror changes height, so it changes the second coordinate.

Answer. $A'=(1,-3)$, $B'=(4,-3)$, $C'=(2,-5)$.

Check. $\frac{1+(-3)}2=-1$ and $\frac{3+(-5)}2=-1$, placing both midpoints on the mirror.

Watch out

Wrong: reflection in $y=-1$ sends $(2,3)$ to $(2,-3)$. That uses the $x$-axis. Right: $3$ is $4$ above $-1$, so its image is $4$ below $-1$, at $-5$.

Reflections in the diagonal mirrors

These rules use the stated lines through the origin. A point on the mirror remains fixed.

Not on the formulae sheet: learn it.

Worked example 3: swapping the coordinates

Question. Reflect $A=(1,2)$, $B=(1,4)$, $C=(2,2)$ in $y=x$. Non-calculator.

Plan. Swap the two coordinates of each vertex.

Why this step? The mirror exchanges the roles of horizontal and vertical position; it does not change both signs.

Answer. $A'=(2,1)$, $B'=(4,1)$, $C'=(2,2)$.

Check. The midpoint of $B=(1,4)$ and $B'=(4,1)$ is $(2.5,2.5)$, on $y=x$; their joining line is perpendicular to the mirror.

Worked example 4: swapping and changing signs

Question. Reflect the same vertices $A=(1,2)$, $B=(1,4)$, $C=(2,2)$ in $y=-x$. Non-calculator.

Plan. Make the new first coordinate the negative of the old second, and the new second the negative of the old first.

Why this step? The line $y=-x$ is a different mirror. The point $(2,2)$, fixed in the previous example, now moves.

Answer. $A'=(-2,-1)$, $B'=(-4,-1)$, $C'=(-2,-2)$.

Check. The midpoint of $(1,4)$ and $(-4,-1)$ is $(-1.5,1.5)$, which lies on $y=-x$.

Watch out

Wrong: reflection in $y=-x$ sends $(1,4)$ to $(-1,-4)$. That is a half-turn about the origin. Right: swap the coordinates as well, giving $(-4,-1)$.

Rotation

A transformation specified by a centre, an angle and a direction, clockwise or anticlockwise. At $180^\circ$, either direction gives the same image. A rotation preserves corresponding lengths and angles.