Transformations & Similarity

How to do it:

1. Slide every point by the same vector $(h,k)\text{.}$

2. Rule: $(x,y) \mapsto (x+h,\, y+k)\text{.}$

3. Check one vertex, then apply the same pair to every vertex.

Worked example: Move right $3$ and down $2\text{:}$ $(x,y) \mapsto (x+3,\, y-2)\text{.}$ The point $(-2,5)$ goes to $(1,3)\text{.}$

Check: every point must move the same direction and distance. Changing only one vertex breaks congruence.

How to do it:

1. Flip across a line.

2. Across the $x$-axis: $(x,y) \mapsto (x,\,-y)\text{.}$ Across the $y$-axis: $(x,y) \mapsto (-x,\, y)\text{.}$ Across $y=x\text{:}$ $(x,y) \mapsto (y,\, x)\text{.}$

3. Sketch the mirror line first so signs do not wander.

Worked example: Reflect $(4,-1)$ across the $x$-axis to $(4,1)\text{.}$ Reflect $(3,2)$ across the $y$-axis to $(-3,2)\text{.}$ A game avatar mirrored across a vertical line uses the $y$-axis rule.

Check: reflecting across the $x$-axis does not negate $x\text{.}$ $(4,-1)$ must not become $(-4,-1)\text{.}$

How to do it:

1. A vertical mirror $x=a$ keeps $y$ and sends $x$ to $2a-x\text{:}$ the image lands as far past the line as the point started before it.

2. A horizontal mirror $y=b$ keeps $x$ and sends $y$ to $2b-y\text{.}$

3. Check with distances: the point and its image are the same distance from the mirror line.

Worked example: reflect $(5,3)$ across $x=2\text{:}$ $x$ becomes $2(2)-5=-1\text{,}$ so the image is $(-1,3)\text{.}$ Reflect $(4,6)$ across $y=-1\text{:}$ $y$ becomes $2(-1)-6=-8\text{,}$ so the image is $(4,-8)\text{.}$

Check: $(5,3)$ is $3$ units right of $x=2\text{,}$ and $(-1,3)$ is $3$ units left of it. The axes are the special cases $a=0$ and $b=0\text{.}$

How to do it:

1. Use counterclockwise about $(0,0)$ unless a problem says otherwise.

2. $90°$ CCW: $(x,y) \mapsto (-y,\, x)\text{.}$ $180°\text{:}$ $(x,y) \mapsto (-x,\,-y)\text{.}$ $270°$ CCW (or $90°$ CW): $(x,y) \mapsto (y,\,-x)\text{.}$

3. Label clockwise versus counterclockwise on the sketch.

Worked example: Rotate $(2,1)$ $90°$ CCW about the origin to $(-1,2)\text{.}$ Rotate $(2,3)$ the same way to $(-3,2)\text{.}$

Check: swapping coordinates without the sign pattern is not a $90°$ turn. Clockwise uses a different rule than counterclockwise.

How to do it:

1. Line up each preimage point with its image and compare coordinates.

2. The same amount added to every $x$ and every $y\text{:}$ a translation. One coordinate flips sign: a reflection over an axis. Coordinates swap with a sign pattern: a rotation of $90°\text{.}$ Both coordinates multiplied by the same number: a dilation.

3. Confirm the rule on a second point before you name the move.

Worked example: $(2,3)\mapsto(5,1)$ and $(0,0)\mapsto(3,-2)\text{:}$ add $3$ to $x$ and $-2$ to $y\text{,}$ a translation by $(3,-2)\text{.}$ $(2,3)\mapsto(-3,2)\text{:}$ the rule $(x,y)\mapsto(-y,x)\text{,}$ a $90°$ counterclockwise rotation. $(2,3)\mapsto(4,6)\text{:}$ a dilation with $k=2\text{.}$

Check: one point is never enough. $(2,3)\mapsto(-2,3)$ could be a reflection over the $y$-axis or a translation by $(-4,0)\text{;}$ a second point settles it.

How to do it:

1. Apply the moves one at a time, in the order given, writing the coordinates after each step.

2. To carry one figure onto another, first match orientation (reflect if the figure is flipped), then rotate if needed, then translate to line up a vertex.

3. Order matters: reflecting then translating usually differs from translating then reflecting.

Worked example: reflect $(1,2)$ across the $y$-axis to get $(-1,2)\text{,}$ then translate by $(0,-5)$ to get $(-1,-3)\text{.}$ In the other order: translate first to $(1,-3)\text{,}$ then reflect to $(-1,-3)\text{,}$ the same result because the translation was vertical. With a horizontal translation by $(3,0)\text{,}$ the two orders give $(2,2)$ and $(-4,2)\text{.}$

Check: any sequence of translations, reflections, and rotations produces a congruent figure. Adding a dilation makes the figures similar instead.

How to do it:

1. Congruent figures match in size and shape; one can be carried onto the other by rigid motions.

2. Similar figures have equal angle measures and proportional side lengths. Scale factor $k$ multiplies every length.

3. Dilation about $(0,0)\text{:}$ $(x,y) \mapsto (kx,\, ky)\text{.}$ $k>1$ enlarges; $0<k<1$ shrinks. Lengths scale by $k\text{;}$ areas scale by $k^2\text{.}$

Worked example: Two sprites that differ only by a mirror flip are congruent. A mini-map icon with $k=3$ is similar, not congruent. A photo pinch to $150\%$ is $k=1.5\text{.}$ Map scale $1$ cm : $5$ km is a dilation relating drawing lengths to real lengths.

Check: order matters when you chain moves: reflect then translate may land differently than translate then reflect. Stretching only width breaks similarity.

How to do it:

1. Rigid motions keep lengths, angle measures, and parallel lines; lines go to lines, and segments go to segments of the same length.

2. Dilations keep angle measures and parallel lines but multiply every length by $k\text{.}$

3. Verify a length with the distance formula, and a right angle by checking the slopes of the sides.

Worked example: a triangle with vertices $(0,0)\text{,}$ $(3,0)\text{,}$ $(0,4)$ has sides $3\text{,}$ $4\text{,}$ $5$ and a right angle at the origin. After a $90°$ rotation the vertices are $(0,0)\text{,}$ $(0,3)\text{,}$ $(-4,0)\text{:}$ still $3\text{,}$ $4\text{,}$ and $\sqrt{16+9}=5\text{.}$ After a dilation by $k=2$ the sides are $6\text{,}$ $8\text{,}$ $10\text{,}$ and the angle is still $90°\text{.}$

Check: if a side length changed after a "rigid motion," a coordinate was copied wrong. Parallel sides stay parallel under every transformation in this topic.