Three ways to move a figure without changing its size or shape.
Find the image
The point A(3, −2) is rotated 90° counterclockwise about the origin. Where does it land?
A 90° counterclockwise turn sends (x, y) to (−y, x), so (3, −2) becomes (2, 3). Sketch it to be sure: A sits in the fourth quadrant, and a quarter turn anticlockwise lands it in the first ✓ C is the clockwise turn, A is a half turn of 180°, and D is a reflection in the x-axis, not a rotation at all.
1
All three moves keep the figure congruent.
Same lengths, same angles, same area — only the position changes.
Transformation
What it does
Needs
Translation
slides
how far across and up
Reflection
flips
a mirror line
Rotation
turns
a centre, an angle, a direction
The original is the object; the result is the image, usually labelled
with primes — A becomes A′.
Because nothing stretches, these are called rigid motions. A transformation that did change the size, such as an enlargement, would not belong here.
2
Translation: add to each coordinate.
Right and up are positive; left and down are negative.
B(−1, 4), 5 right and 3 down
the instruction
(x + 5, y − 3)
the rule
B′(4, 1)
−1 + 5 and 4 − 3
Every point of the figure moves by the same amount, so a translated triangle is
never turned or flipped — it faces exactly the way it did.
3
Reflection: one coordinate changes sign.
Mirror line
Rule
P(4, 3) becomes
the x-axis
(x, −y)
(4, −3)
the y-axis
(−x, y)
(−4, 3)
the line y = x
(y, x)
(3, 4)
the origin
(−x, −y)
(−4, −3)
Which axis flips which
Reflecting in the x-axis changes the sign of y,
not x. The point moves across the mirror, and the mirror here is horizontal, so
the vertical coordinate is the one that flips.
4
Rotation about the origin: swap and change one sign.
90° counterclockwise: (x, y) → (−y, x)
Turn
Rule
(3, −2) becomes
90° counterclockwise
(−y, x)
(2, 3)
180° (either way)
(−x, −y)
(−3, 2)
270° counterclockwise
(y, −x)
(−2, −3)
90° clockwise
(y, −x)
(−2, −3)
The last two rows are the same rule, because turning 270° anticlockwise and 90°
clockwise land you in the same place. At 180° the direction stops mattering.
If the rules will not stay in your head, sketch the point and turn the page. A quarter turn from the fourth quadrant reaches the first — that alone eliminates most wrong choices.
5
Dilation: the one move that changes the size.
Dilation about the origin by scale factor k: (x, y) → (kx, ky). Lengths × k, area × k².
A square with vertex Q(−2, 3) and area 36 cm² is dilated about the origin by a scale factor of 12. Find Q′ and the new area.
Q′ = ((12)(−2), (12)(3)) = (−1, 1.5)
multiply both coordinates
area × (12)² = 36 × 14
area scales by k²
= 9 cm²
side 6 cm became 3 cm: 3 × 3 = 9 ✓
Area does not scale by k
Halving the area gives 18 cm², but each side is halved, and the area is side × side, so it is quartered. A dilation is not a rigid motion: shape and angles are kept, but lengths and area are not.
6
Variations: a clockwise turn, the line y = −x, a mirror line off the axes.
The same rules; each case is one the tables above do not list directly.
270° clockwise: rotate (−4, 1) about the origin
270° clockwise = 90° counterclockwise
360° − 270° = 90° the other way
(x, y) → (−y, x) gives (−1, −4)
second quadrant to third ✓
Different: change a large turn into the equal quarter turn the other way.
The line y = −x: reflect (3, −5)
(x, y) → (−y, −x)
swap AND change both signs
(3, −5) → (5, −3)
check: the midpoint (4, −4) lies on y = −x ✓
Different: y = x only swaps; y = −x swaps and negates.
A mirror line that is not an axis: reflect (5, 1) in x = 2
5 − 2 = 3
3 units right of the mirror
2 − 3 = −1, so the image is (−1, 1)
3 units left of it; y unchanged
Different: (−x, y) only works for the y-axis. For any other line, measure the distance to the mirror and go the same distance past it.
7
One after another, and working backwards.
C(2, 5), reflect in the x-axis
→ (2, −5)
then translate 3 left, 1 up
→ C′(−1, −4)
Backwardsimage D′(6, −1), rule (x + 4, y − 3)find the object
Undo it(6 − 4, −1 + 3)D(2, 2) ✓
Remember the flow
Name the transformation · Apply its coordinate rule · One point at a time · Sketch to check the quadrant
Order matters: reflect-then-translate rarely gives the same image as translate-then-reflect.
On the exam
Expect the image of one point under a named move, the name of the move that maps one figure onto another, and a two-step or backwards item. The fastest route is the coordinate rule for each move, applied one step at a time. The 10-second check: sketch the point and see whether the image lands in the quadrant the move should send it to.
Now you try
Practice problem
Name the transformation
A triangle has vertices (1, 2), (4, 2) and (4, 6). Its image has vertices (2, 1), (2, 4) and (6, 4). Which transformation was applied?
Every point has had its coordinates swapped: (1, 2) → (2, 1), (4, 2) → (2, 4), (4, 6) → (6, 4). That is the rule (x, y) → (y, x), which is a reflection in the line y = x. B would have made the y-values negative, and C would have made both negative. A translation adds the same amount to every point, which is not what happened here.
Hint: compare one vertex with its image first and guess the rule,
then test that rule on the other two. A rule that fits all three is the answer.
