Translate, reflect, and rotate shapes on a Cartesian plane

Ontario Grade 7 Mathematics

Spatial Sense

Translations, reflections, and rotations

A shape on a Cartesian plane is made from points with locations we can name. We can move the shape without changing its size or shape. In this lesson, you will learn three ways to move it: translate it, reflect it, and rotate it. We will keep track of each vertex so that the new shape is in the right place.

What you will learn

Read and name points on a Cartesian plane.
Describe how a translation moves every point the same distance in the same direction.
Reflect a shape across an axis by changing the sign of one coordinate.
Rotate a shape one quarter-turn around the origin by changing and switching its coordinates.

Read the plane and name the points

A Cartesian plane has two number lines that cross. The horizontal line is the x-axis. The vertical line is the y-axis. Their crossing point is the origin, written

(0,0)

. A coordinate is a pair of numbers that names a point.

In an ordered pair

(x,y)

, the first number tells how far to move horizontally from the origin. The second tells how far to move vertically. Positive x goes right, and negative x goes left. Positive y goes up, and negative y goes down.

A vertex is a corner of a shape. To move a shape, move every vertex using the same rule. Then connect the new vertices in the same order. The moved shape has the same size and shape as the starting shape.

Read the x-coordinate first and the y-coordinate second.
The origin is where the axes meet.
Keep the vertices in order when you draw the shape.

Translate: slide every point

A translation is a slide. Imagine a paper shape on a desk. If you slide it three spaces right and two spaces up, every corner moves three spaces right and two spaces up. The shape does not turn or flip.

For a translation, add the horizontal move to every x-coordinate and the vertical move to every y-coordinate. For example, a move four units left and one unit down changes each point by the same amounts. A unit here means one marked step on the grid.

Check your work by comparing each starting point with its new point. The horizontal and vertical changes must be the same for every vertex.

(x,y)\rightarrow(x+a,y+b)

A translation slides without turning or flipping.
Right and up increase coordinates; left and down decrease them.
Apply the same horizontal and vertical changes to every vertex.

Reflect: flip across a line

A reflection is a flip across a line. The line is called the line of reflection. Each point and its image sit the same distance from that line, on opposite sides. An image is the new point or shape after a move.

When a shape is reflected across the x-axis, its horizontal position stays the same, but its vertical position changes to the opposite sign. A point above the x-axis moves the same distance below it. When a shape is reflected across the y-axis, the vertical position stays the same, but the horizontal position changes sign.

For example, reflecting

(2,3)

across the x-axis gives

(2,-3)

. Reflecting that same starting point across the y-axis gives

(-2,3)

. The unchanged coordinate tells you the point stays the same distance from the line of reflection.

(x,y)\xrightarrow{\text{x-axis}}(x,-y),\quad (x,y)\xrightarrow{\text{y-axis}}(-x,y)

Across the x-axis, keep x and change the sign of y.
Across the y-axis, change the sign of x and keep y.
A point on the line of reflection stays in place.

Rotate: turn around the origin

A rotation is a turn around a fixed point. In this lesson, the fixed point is the origin. A quarter-turn is one of four equal turns that make a full turn. We will use quarter-turns in either direction.

For a quarter-turn counterclockwise, each point moves from its starting place around the origin. Its coordinates switch places, and the original x-coordinate changes sign. For a quarter-turn clockwise, the coordinates also switch places, but the original y-coordinate changes sign.

The direction matters. Counterclockwise means the direction a clock hand does not travel; clockwise means the direction a clock hand travels. Keep the same turn direction for every vertex. A point at the origin stays there because it is the centre of the turn.

(x,y)\xrightarrow{90^\circ\text{ counterclockwise}}(-y,x),\quad (x,y)\xrightarrow{90^\circ\text{ clockwise}}(y,-x)

Name the direction of the turn.
For a quarter-turn, switch the coordinates and change one sign.
Turn every vertex around the origin by the same amount.

Plan, draw, and check

Start by listing the original vertices in order. State the move clearly: the direction and distance for a translation, the axis for a reflection, or the direction and amount for a rotation. Apply the rule to each vertex before drawing.

Plot the new points by moving horizontally first and vertically second. Connect the points in the original order. Label the new vertices so you can compare them with the starting shape.

Check that the result matches the stated move. A translation should have the same coordinate changes at every point. A reflection should place matching points equally far from its line. A rotation should turn points around the stated centre in the stated direction. If one corner does not follow the rule, revisit that coordinate.

Write down all starting vertices before moving the shape.
Use one rule consistently for every vertex.
Check the direction, axis, or centre before deciding the answer is correct.

Worked example

Slide a triangle

Triangle ABC has vertices

A(1,1)

,

B(4,1)

, and

C(2,3)

. Translate it 2 units left and 3 units up. Find the new vertices.

Translated triangle

The plotted vertices are the translated triangle: A′(-1,4), B′(2,4), and C′(0,6), connected in order.

Describe the move
Moving left reduces every x-coordinate by 2. Moving up increases every y-coordinate by 3. Use those same changes for all three vertices.
$(x,y)\rightarrow(x-2,y+3)$
Move each vertex
For A, subtract 2 from 1 and add 3 to 1. For B, subtract 2 from 4 and add 3 to 1. For C, subtract 2 from 2 and add 3 to 3.
$(1,1)\rightarrow(-1,4),\quad(4,1)\rightarrow(2,4),\quad(2,3)\rightarrow(0,6)$
State the result
The new triangle has vertices A′, B′, and C′ at the listed points. Each vertex moved left 2 and up 3, so the translation is consistent.

Answer: The new vertices are

A′(-1,4)

,

B′(2,4)

, and

C′(0,6)

.

Check: Each new x-coordinate is 2 less than before, and each new y-coordinate is 3 greater.

Worked example

Reflect a quadrilateral

Quadrilateral PQRS has vertices

P(-3,1)

,

Q(-1,1)

,

R(-1,4)

, and

S(-3,3)

. Reflect it across the y-axis. Find the image vertices.

Reflected quadrilateral

The plotted image vertices are P′(3,1), Q′(1,1), R′(1,4), and S′(3,3), connected in order.

Choose the reflection rule
A reflection across the y-axis keeps each y-coordinate and changes the sign of each x-coordinate. The shape moves to the other side of the y-axis.
$(x,y)\rightarrow(-x,y)$
Reflect each vertex
Change the sign of the first coordinate in each ordered pair. Keep the second coordinate unchanged.
$P(-3,1)\rightarrow P′(3,1),\quad Q(-1,1)\rightarrow Q′(1,1),\quad R(-1,4)\rightarrow R′(1,4),\quad S(-3,3)\rightarrow S′(3,3)$
Check the placement
Each original point and its image have opposite x-coordinates and equal y-coordinates. Each pair is the same distance from the y-axis.

Answer: The image vertices are

P′(3,1)

,

Q′(1,1)

,

R′(1,4)

, and

S′(3,3)

.

Check: The y-coordinates stay the same, while the x-coordinates change sign.

Worked example

Turn a triangle one quarter-turn

Triangle DEF has vertices

D(2,1)

,

E(4,1)

, and

F(3,3)

. Rotate it 90° counterclockwise around the origin. Find the image vertices.

Rotated triangle

The plotted image vertices are D′(-1,2), E′(-1,4), and F′(-3,3), connected in order.

Select the turn rule
For a 90° counterclockwise turn around the origin, switch the coordinates and change the sign of the original x-coordinate.
$(x,y)\rightarrow(-y,x)$
Rotate each vertex
For D, the new coordinates are the opposite of 1 and then 2. Apply the same coordinate change to E and F.
$(2,1)\rightarrow(-1,2),\quad(4,1)\rightarrow(-1,4),\quad(3,3)\rightarrow(-3,3)$
Check the direction
The original points are to the right of the origin. After a counterclockwise quarter-turn, their images are above or to the left of the origin, as shown by the new coordinates.

Answer: The image vertices are

D′(-1,2)

,

E′(-1,4)

, and

F′(-3,3)

.

Check: For each point, the coordinates switch places and the original x-coordinate becomes negative.

Common mistakes and how to avoid them

Changing the x-coordinate when moving up or down.

Correction: Up and down change the y-coordinate. Left and right change the x-coordinate.

Changing both coordinates in a reflection across one axis.

Correction: Across the x-axis, only y changes sign. Across the y-axis, only x changes sign.

Using the clockwise rule for a counterclockwise turn.

Correction: Name the turn direction first, then use its matching coordinate rule.

Moving only one or two vertices of a shape.

Correction: Apply the same stated move to every vertex, then connect the new points in order.

Lesson summary

A point is named by its horizontal coordinate first and its vertical coordinate second.
A translation slides every point by the same horizontal and vertical amounts.
A reflection across an axis changes the sign of the coordinate perpendicular to that axis.
A quarter-turn around the origin switches coordinates and changes one sign; the direction tells you which sign changes.

Check your understanding

Question 1

A point

(3,-2)

is translated 1 unit left and 4 units up. What is its new location?

$(2,2)$
$(4,2)$
$(2,-6)$
$(3,2)$

Show answer and explanation

(2,2)

Subtract 1 from x and add 4 to y. The result is

(2,2)

.

Question 2

What is the reflection of

(-5,2)

across the x-axis?

$(5,2)$
$(-5,-2)$
$(5,-2)$
$(-5,2)$

Show answer and explanation

(-5,-2)

A reflection across the x-axis keeps x and changes the sign of y.

Question 3

A point

(1,4)

is rotated 90° clockwise around the origin. Where does it move?

$(-4,1)$
$(4,-1)$
$(4,1)$
$(-1,-4)$

Show answer and explanation

(4,-1)

For a clockwise quarter-turn, switch the coordinates and change the sign of the original y-coordinate, giving

(4,-1)

.

Key terms

Cartesian plane: A plane with a horizontal number line and a vertical number line used to locate points.
Coordinate: A number in an ordered pair that helps name a point's location.
Vertex: A corner of a shape.
Translation: A slide that moves every point the same distance in the same direction.
Reflection: A flip across a line.
Rotation: A turn around a fixed point.
Origin: The point where the x-axis and y-axis meet, written $(0,0)$ .

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Published by DoAssignment. This reviewed lesson follows Ontario Grade 7 Mathematics, expectation E1.4. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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E1.4 · Translate, reflect, and rotate shapes on a Cartesian plane

Ontario Grade 7 Mathematics

Translations, reflections, and rotations

What you will learn

Read the plane and name the points

Translate: slide every point

Reflect: flip across a line

Rotate: turn around the origin

Plan, draw, and check

Slide a triangle

Reflect a quadrilateral

Turn a triangle one quarter-turn

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through Grade 7 Math

About this lesson and its review