A translation adds a fixed amount to each coordinate.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Reflection
Across the y-axis, (x, y) becomes (−x, y). Across the x-axis, (x, −y).
Rotation about the origin
A quarter turn anticlockwise sends (x, y) to (−y, x).
All three are rigid
Lengths and angles survive every one of them, which is why the image is congruent.
Each motion has a coordinate rule
Translation adds constants: (x, y) → (x + a, y + b). Reflection in the x-axis negates y. A 90° anticlockwise rotation about the origin sends (x, y) to (−y, x). The rules can be discovered by plotting one point and turning the page.
Composing rules
Applying two transformations means applying one rule to the output of the other. Order matters in general, so the composition must be written and applied in the stated sequence rather than combined by intuition.
Fixed points
A rotation about the origin fixes only the origin. A reflection fixes every point on the mirror line. A nontrivial translation fixes nothing. Identifying fixed points is often the quickest way to recognise which transformation was applied.
Where this leads
These rules are exactly what matrices encode, which is how graphics hardware moves millions of points per frame. Coordinate transformations are the bridge from school geometry to computer graphics and linear algebra.
Step 2: Try It Yourself
Tap and try it out.
Every length and every angle is exactly as it was. The shape moved without changing, so the two figures are congruent.
Step 3: Watch an Example
One step at a time.
Watch Kofi Rotate a Corner
A corner at (3, 1) is rotated a quarter turn anticlockwise about the origin.
- Step 1
The rule sends (x, y) to (−y, x).
Step 4: Your Turn
Practice makes it stick.
Reflecting
Problem 1 of 2
(5, 2) reflected across the y-axis. What is the new x?
Translating
Problem 2 of 2
(4, 7) translated 3 left. What is the new x?
Apply the Rule
1 of 8
(2, 6) reflected across the x-axis. New y?
2 of 8
(−3, 4) reflected across the y-axis. New x?
3 of 8
(1, 5) translated 4 up. New y?
4 of 8
(2, 3) rotated 90° anticlockwise. New x?
5 of 8
Do rigid motions change side lengths? 1 for yes, 0 for no.
6 of 8
Match each motion to its coordinate rule.
Tap a card on the left to start.
7 of 8
(0, 4) rotated 180° about the origin. New y?
8 of 8
(6, 6) reflected across the line y = x. New x?
Step 5: Quick Check
Show what you know.
Question 1 of 1
(7, 2) reflected across the x-axis. New y?
What You Learned
- Translations add, reflections flip one sign, and a quarter turn swaps the coordinates.
- All three preserve lengths and angles.
- That preservation is exactly why the image is congruent to the original.