Translations, Reflections & Rotations — Print

Three moves — slide, flip, turn — never change a shape. Everything in this unit follows from that one fact.

What the practice questions cover

These questions use 8 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.

Vertex — A flat surface; a line where two faces meet; a corner point.
Image — The shape after the move. Its corners get primes: A', B', C'.
Rigid motion — A move that keeps every length and every angle. Slides, flips and turns are all rigid.
Translation — A slide. Every point moves the same distance in the same direction.
Reflection — A flip across a line, like a mirror.
Rotation — A turn about a fixed point, through a given angle and direction.
Congruent — Same size and same shape. Written △ ABC ≅ △ DEF.
Transformation — A change to the function that moves or stretches its graph.

Cut a triangle out of card. Now slide it across the desk, flip it over, and spin it round.

Nothing you just did changed the card. The same three edges are still the same three lengths.

The dot marks one corner, so you can follow where that corner went.

After the flip the dot is on the other side — the card is face down. After the spin it has swung round.

But measure any edge at any moment and you get the same answer. The card was carried, never stretched.

That is the whole of this unit. Slides, flips and turns are called rigid motions because the shape rides along unchanged — so everything that follows is just bookkeeping about where each corner ended up.

What stays the same

A slide, a flip and a turn all leave the shape exactly as big and exactly as bent as it was. Side lengths do not change. Angles do not change. Only the position changes.

That is why they are called rigid motions — imagine the shape cut from card. You can move the card around, but you cannot stretch it.

Everything else in this unit is bookkeeping: which point went where.

The one thing to remember

After a rigid motion, the image is congruent to the pre-image. Same size, same shape, new position.

Translations: slide

A translation is described by how far right or left and how far up or down. "5 right, 2 down" means add 5 to every x and subtract 2 from every y.

In symbols: (x, y) → (x + 5, y - 2).

Every single point moves by the same amount. If one corner goes 5 right and 2 down, so does every other corner.

Do it one corner at a time

Write the three corners in a column. Do the x for all three, then the y for all three. Doing one whole point at a time is where mixed-up coordinates come from.

Reflections: flip

A reflection across the x-axis keeps x and flips the sign of y: (x, y) → (x, -y).

A reflection across the y-axis keeps y and flips the sign of x: (x, y) → (-x, y).

Every point ends up the same distance from the mirror line, but on the other side. A point sitting on the mirror line does not move at all.

Which sign flips?

Across the x-axis, the y flips. Across the y-axis, the x flips. It feels backwards, so say it out loud: "over the x-axis, y changes."

Rotations: turn

A rotation needs three things: a centre, an angle, and a direction. Almost all school rotations are about the origin, by 90°, 180° or 270°.

About the origin, counter-clockwise: 90° gives (x, y) → (-y, x); 180° gives (x, y) → (-x, -y); 270° gives (x, y) → (y, -x).

If you forget, turn the paper. Physically rotating the page a quarter turn and reading off the new coordinates always works.

A trick that never fails

For 90° counter-clockwise: swap the two numbers, then flip the sign of the new first one. (3, 5) → (5, 3) → (-5, 3).

Describing a move you are shown

Given a pre-image and an image, work out which move happened by comparing one matching pair of points.

Same orientation, just shifted? Translation. Mirror-flipped? Reflection. Spun round? Rotation.

Then check a second pair of points to make sure. One pair can look like two different moves; two pairs almost never can.

Worked examples

Translating a triangle

Triangle ABC has A(1, 2), B(4, 2), C(4, 6). Translate it 3 left and 5 up.

Write the rule: (x, y) → (x - 3, y + 5) Left means subtract from x; up means add to y.
A(1, 2) → A'(-2, 7) 1 - 3 = -2 and 2 + 5 = 7.
B(4, 2) → B'(1, 7) Same rule, same amounts.
C(4, 6) → C'(1, 11) Check: A'B' is still 3 long, just as AB was.

A'(-2, 7), B'(1, 7), C'(1, 11)

Reflecting across an axis

Reflect the point P(-6, 3) across the x-axis.

The mirror is the x-axis, so the y changes sign. Distance from the x-axis is measured by y.
(-6, 3) → (-6, -3) P was 3 above the axis; P' is 3 below it.

P'(-6, -3)

Rotating about the origin

Rotate Q(2, -5) by 90° counter-clockwise about the origin.

Rule: (x, y) → (-y, x) Swap, then negate the new first coordinate.
Swap: (2, -5) → (-5, 2) The y value moves to the front.
Negate the front: (5, 2) -(-5) = 5.
Check the distance from the origin. √2² + 5² before and √5² + 2² after — equal, as a rotation demands.

Q'(5, 2)

Words to know

Pre-image: The shape you start with. Its corners are labelled A, B, C.
Image: The shape after the move. Its corners get primes: A', B', C'.
Rigid motion: A move that keeps every length and every angle. Slides, flips and turns are all rigid.
Translation: A slide. Every point moves the same distance in the same direction.
Reflection: A flip across a line, like a mirror.
Rotation: A turn about a fixed point, through a given angle and direction.

Mistakes to watch for

Flipping the wrong coordinate. Across the x-axis it is the y that changes sign, because y measures how far you are from the x-axis. (6, -4) → (6, 4). The name and the letter that moves are deliberately mismatched, so say it out loud each time.

Translating one corner and drawing the rest by eye. A translation moves every point by the same amount, so if A goes 5 right then B and C do too. Doing one and guessing the others is how a triangle quietly changes shape.

Mixing up clockwise and counter-clockwise. Unless a question says otherwise, rotations are counter-clockwise — the direction angles are measured in. A 90° clockwise turn is the same as 270° counter-clockwise, which is why both labels appear on the same picture.

Expecting a rotation to change the size. It never does. If your image looks bigger, you have dilated it by accident — check that the distance from the origin to a corner is the same before and after.

The word congruent comes from Latin congruere, "to come together, to agree." Two shapes that agree in every measurement are congruent.

Euclid wrote about these motions around 300 BC, but he described them as physically picking a figure up and laying it down on another.

Treating them as rules on coordinates is much newer — it needed Descartes to invent the coordinate plane in the 1630s, nearly two thousand years later.

Problem 1

Translate A(3, 5) by the rule (x, y) → (x + 4, y - 2).

Worked answer

(7, 3)

3 + 4 = 7

5 - 2 = 3

A'(7, 3)

Problem 2

Translate B(-2, 7) five units left and three units down.

Worked answer

(-7, 4)

-2 - 5 = -7

7 - 3 = 4

B'(-7, 4)

Problem 3

Reflect P(6, -4) across the x-axis.

Worked answer

(6, -(-4)) = (6, 4)

x stays 6

-4 becomes 4

P'(6, 4)

Problem 4

Reflect Q(-3, 8) across the y-axis.

Worked answer

(3, 8)

-3 becomes 3

y stays 8

Q'(3, 8)

Problem 5

Rotate R(4, 1) by 180° about the origin.

Worked answer

(-4, -1)

Rule: (x, y) → (-x, -y)

R'(-4, -1)

Problem 6

Rotate S(2, 7) by 90° counter-clockwise about the origin.

Worked answer

(-7, 2)

Swap: (7, 2)

Negate the front: (-7, 2)

S'(-7, 2)

Problem 7

Rotate T(-5, 3) by 270° counter-clockwise about the origin.

Worked answer

(3, 5)

y = 3 moves to the front

-x = 5

T'(3, 5)

Problem 8

A triangle has vertices (1, 1), (4, 1), (4, 5). After a translation, the image of (1, 1) is (-2, 6). Where does (4, 5) go?

Worked answer

(1, 10)

1 → -2 means -3 on x

1 → 6 means +5 on y

(4 - 3, 5 + 5) = (1, 10)

Problem 9

Point (5, -2) maps to (-5, -2). Which single transformation was used?

Worked answer

reflection across the y-axis

The x changed sign; the y did not

That is the rule (x, y) → (-x, y)

A reflection across the y-axis

Problem 10

A figure is moved so that every point travels 6 units right and 0 units vertically. Which is true of the image?

It is congruent to the original
It is larger than the original
It is a mirror image of the original
It has different angle measures

Worked answer

It is congruent to the original

A translation changes position only

Lengths and angles are unchanged

So the image is congruent

Problem 11

Reflect (a, b) across the x-axis, then across the y-axis. Give the result.

Worked answer

(-a, -b)

Across the x-axis: (a, -b)

Across the y-axis: (-a, -b)

Same as a 180° rotation

Problem 12

Point M(2, -3) is rotated 90° counter-clockwise about the origin, then translated 1 unit up. Find the image.

Worked answer

(3, 3)

Rotate: (2, -3) → (3, 2)

Translate up 1: (3, 3)

Problem 13

A point is reflected across the y-axis and then translated 4 right. The final image is (1, 6). Find the original point.

Worked answer

(3, 6)

Undo the translation: (1 - 4, 6) = (-3, 6)

Undo the reflection: (3, 6)

Check: (3,6) → (-3,6) → (1,6) ✓

Problem 14

How many degrees counter-clockwise about the origin will map (4, 0) to (0, -4)?

Worked answer

270

90° takes it to (0, 4)

180° to (-4, 0)

270° to (0, -4)

Problem 15

Triangle ABC has A(1, 2), B(5, 2), C(1, 6). Its image is A'(2, -1), B'(2, -5), C'(6, -1). Which transformation was used?

Worked answer

rotation of 90 degrees clockwise about the origin

A(1,2) → (2, -1) ✓

B(5,2) → (2, -5) ✓

C(1,6) → (6, -1) ✓

That rule is a 90° clockwise rotation, also called 270° counter-clockwise

Problem 16

A translation of 3 right and 2 up is followed by another of 1 left and 5 down. Write the single rule that does both.

Worked answer

(x + 2, y - 3)

Horizontal: +3 - 1 = +2

Vertical: +2 - 5 = -3

(x, y) → (x + 2, y - 3)

Browse all courses