Slide, flip, turn, resize. Move one corner by the rule and the rest follow. If only slides, flips and turns connect two figures, they are congruent; if it also takes a resize, they are similar — and naming the moves is the proof. With parallel lines, a transversal, and why a triangle’s angles make 180°. Study cards, hints, a practice quiz at three levels, and four workshop activities that check themselves — one on a four-quadrant plane where you flip a triangle corner by corner.
One triangle. Every corner has an address — an x and a y.
Slides, flips and turns keep size and shape — that makes the figures congruent. Add a resize and they are similar instead. Naming the moves IS the proof.
Four things people get wrong
⚠️People often think…
Reflecting over the x-axis changes the x.
The axis you flip over is the coordinate that STAYS. Flip over the x-axis and the y changes sign; flip over the y-axis and the x changes sign. It feels backwards, and it is not.
The axis named is the one that survives.
⚠️People often think…
A dilation means adding the same number to each coordinate.
A dilation MULTIPLIES both coordinates — that is what makes it bigger or smaller. Adding the same number to each is a translation, which just slides it.
Multiply to resize. Add to slide.
⚠️People often think…
A 90° turn and a 180° turn work about the same way.
For 90°, the coordinates SWAP and one sign changes. For 180°, nothing swaps and BOTH signs change. Two different moves with two different rules.
Quarter turn: swap. Half turn: both signs.
⚠️People often think…
If you enlarge a shape by 3, the angles get bigger too.
Angles never scale. A 60° corner is still 60° in the enlargement — that is exactly why the shape still looks like itself. Only the side lengths multiply.
Sides stretch. Corners do not.
Move one corner by the rule
1Watch one
Reflect the point (3, 5) over the x-axis.
The x-axis is the mirror line.
The axis you flip over stays put.
So x stays 3, and y changes sign.
(3, −5).
2Do one with me
Translate the point (−2, 6) right 5 and down 4.
Right and left change which coordinate?
New x:
New y:
💬One sentence, then you move on
Why does flipping over the x-axis change the y and not the x?
3Try one
A shape with a 40° corner is dilated by a factor of 3. What is that corner now? Write the number of degrees.
I want a hint first
Blow a photo up on a copier. The sides get longer — but does a square corner stop being square?
💬Last one — then you're done here
What is the difference between congruent and similar?
Where this goes
Where this lives
Resizing a photo without squashing it, a blueprint drawn to scale, and the pinch-to-zoom on a phone — all of it is a dilation.
What this feeds
Next unit is the Pythagorean theorem — three squares and one equation.
Name one thing you have resized on a screen this week.
One card at a time — tap “Show me” to check yourself, then Next. Start at Foundation; when those feel easy, climb.
Helpful Hints
🪜 From a move to a proof — the ladder
Name the move (slide, flip, turn, resize) → apply its rule to EVERY corner → label the image with primes → ask what stayed the same → only rigid motions: congruent → a dilation in the chain: similar → similar figures: equal angles, sides times one scale factor.
Move one corner right, and the rest follow the same rule.
🤝 The whole topic in one idea
Congruent and similar are not about how figures LOOK — they are about what moves connect them. If slides, flips and turns can carry one onto the other, they are congruent. If it also takes a resize, they are similar. Naming the moves IS the proof.
⚠️ Traps the test loves
Reflecting over the X-axis changes Y. The axis you flip over is the coordinate that stays.
Left and right change x; up and down change y.
A dilation MULTIPLIES both coordinates. Adding is a translation.
For 90° turns the coordinates swap AND one sign changes. For 180° both signs change and nothing swaps.
Angles do not scale. A 60° angle is 60° in the enlargement.
Similar sides multiply — never add the same amount to each.
📐 The coordinate rules — the chart
Move
Rule
(3, 1) goes to
Rigid?
Translate right a, up b
(x + a, y + b)
right 2, up 4 → (5, 5)
yes
Reflect over the x-axis
(x, −y)
(3, −1)
yes
Reflect over the y-axis
(−x, y)
(−3, 1)
yes
Rotate 90° counterclockwise
(−y, x)
(−1, 3)
yes
Rotate 180°
(−x, −y)
(−3, −1)
yes
Dilate by k from the origin
(kx, ky)
k = 2 → (6, 2)
no
🎯 How the test will ask
A figure and a move — “give the coordinates of the image.”
Two figures on a grid — “describe a sequence that maps one onto the other.”
“Are they congruent or similar? How do you know?” — answer with the moves.
Similar figures with a missing side — find the scale factor first.
Parallel lines and a transversal — find an angle and name the pair.
Two triangles with two angles each — “are they similar?” (find the third angles).
✅ Before the test, can you…
Tell a slide from a flip from a turn from a resize by looking?
Apply each coordinate rule to a point?
Carry a figure through a sequence of two moves?
Explain congruent and similar in terms of which moves are allowed?
Find a missing side in similar figures with a scale factor?
Name the angle pairs made by parallel lines, and use them to explain 180°?
Pick your level
Look back at anything you missed — the hint that appeared is exactly what to reread tonight.
How sure did you feel?
Workshop
Work like a mathematician: name the move, flip a triangle corner by corner on a four-quadrant plane, follow one point through three moves, then decide whether two rectangles are really similar. Every activity checks itself, and hints are free.
Your practice never leaves this device. There is no account and no sign-in. Your work is saved in this browser only, and you can erase it whenever you want.
Your practice record — saved on this device
This is your record of the module on screen — it stays here and goes nowhere. Independent means you got it right on the first tap; supported means you got it after the explain-and-retry, or marked ‘I had it’ on a revealed answer. Both count, and neither is a grade. If your teacher asks, copy the row or show them this screen.
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The answer key is for a teacher: it prints only from here, for the unit on screen. Print the study packet prints the study pages and a blank quiz — never the answers.