Translate, reflect, rotate, and dilate figures on a Cartesian plane

Ontario Grade 8 Mathematics

Spatial Sense

Translations, reflections, rotations, and dilations

A Cartesian plane is a grid made by a horizontal number line and a vertical number line. The horizontal line is the

x

-axis, and the vertical line is the

y

-axis. They meet at the origin,

(0,0)

. A point is named by an ordered pair, such as

(2,3)

. The first number tells how far to move horizontally; the second tells how far to move vertically. A figure is a shape made from points joined by line segments. In this lesson, you will move or resize figures by changing the positions of their points. The new figure is called the image. The figure before the change is called the original.

What you will learn

Describe how a figure changes position or size on a Cartesian plane.
Apply a translation, reflection, rotation, or dilation to points in a figure.
Record the image coordinates and compare them with the original coordinates.

Read points and plan a move

Before transforming a figure, name each corner with a letter and record its coordinates. For example, a triangle might have corners

A(1,1)

,

B(4,1)

, and

C(2,3)

. Plot the points and join them in order. Each corner must receive the same kind of move so the whole triangle changes together.

A translation slides every point the same distance in the same direction. To move a point right, increase its

x

-coordinate. To move it left, decrease that coordinate. Moving up increases the

y

-coordinate; moving down decreases it. For a move right 2 units and up 1 unit, add 2 to every first coordinate and 1 to every second coordinate.

Compare matching points to check your work. The move from

A

to its image must match the move from

B

to its image. If the moves differ, the points were not translated consistently.

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

Record coordinates in the order $(x,y)$ .
A translation moves every point in the same way.
Join image points in the same order as the original points.

Reflect and rotate a figure

A reflection flips a figure across a line called the line of reflection. Imagine folding the plane along that line: matching points would meet. For a reflection across the

x

-axis, keep each

x

-coordinate and change the sign of the

y

-coordinate. For a reflection across the

y

-axis, change the sign of the

x

-coordinate and keep the

y

-coordinate.

A rotation turns a figure around a fixed point called the centre of rotation. Unless another centre is stated, the examples here use the origin. A quarter-turn is a turn of

90^\circ

. A half-turn is a turn of

180^\circ

. For a quarter-turn counterclockwise around the origin, a point

(x,y)

moves to

(-y,x)

. Clockwise means turning in the direction of a clock’s hands; counterclockwise means turning the other way.

After reflecting or rotating, plot each new point and reconnect the corners in their original order. The image may face a different direction, but the points still belong to the same figure.

(x,y)\rightarrow(-y,x)

A reflection flips across a stated line.
A rotation turns around a stated centre through a stated angle and direction.
Check the line or centre before changing coordinates.

Dilate a figure and check the result

A dilation enlarges or reduces a figure from a centre. The scale factor tells how the distances from the centre change. A scale factor greater than 1 makes the figure larger. A scale factor between 0 and 1 makes it smaller. In this lesson, the centre is the origin. Multiply both coordinates of each point by the scale factor. For example, a scale factor of 2 sends

(1,3)

to

(2,6)

.

A dilation changes the size but keeps the figure’s shape. Translations, reflections, and rotations do not change the size. Label new points clearly, such as A', read as “A prime.” The prime mark distinguishes an image point from its original point.

To check your work, confirm that you followed the stated direction, line, centre, angle, or scale factor. Then check every vertex. A missed sign or coordinate can place a point incorrectly.

(x,y)\rightarrow(kx,ky)

A dilation uses a centre and a scale factor.
For a dilation centred at the origin, multiply both coordinates by the scale factor.
Use matching labels to compare the original and image.

Worked example

Translate a triangle

Triangle

ABC

has vertices

A(1,1)

,

B(4,1)

, and

C(2,3)

. Translate it 2 units right and 1 unit up. Find and plot A', B', and C'.

Triangle before and after translation

Plot and connect each set of vertices. Each image point is 2 units right and 1 unit up from its matching original point.

Describe the move
Moving right adds 2 to each $x$ -coordinate. Moving up adds 1 to each $y$ -coordinate. Apply both changes to every vertex.
$(x,y)\rightarrow(x+2,y+1)$
Find the image points
For $A(1,1)$ , the new coordinates are $(3,2)$ . For $B(4,1)$ , they are $(6,2)$ . For $C(2,3)$ , they are $(4,4)$ .
$A'(3,2),\ B'(6,2),\ C'(4,4)$
Check the translation
Each point moved 2 units right and 1 unit up. Plot the three image points and join them in the order A', B', C'.

Answer: The image has vertices

A'(3,2)

,

B'(6,2)

, and

C'(4,4)

.

Check: Each image coordinate is 2 more horizontally and 1 more vertically than its matching original coordinate.

Worked example

Reflect, then rotate a point

Point

P(3,2)

is first reflected across the

y

-axis. Its image is then rotated

90^\circ

counterclockwise about the origin. Find the final point.

Point through two transformations

The labels show the original point, its reflection, and its final rotated position.

Reflect across the y-axis
A reflection across the $y$ -axis changes the sign of the $x$ -coordinate and keeps the $y$ -coordinate. So the first image is $(-3,2)$ .
$(x,y)\rightarrow(-x,y)$
Rotate the reflected point
Use the first image, $(-3,2)$ , as the point to rotate. A $90^\circ$ counterclockwise turn sends $(x,y)$ to $(-y,x)$ . Substituting $x=-3$ and $y=2$ gives $(-2,-3)$ .
$(-3,2)\rightarrow(-2,-3)$
Check the order
The instructions say to reflect first and rotate second. The rotation must therefore start from $(-3,2)$ , not from the original point.

Answer: The final point is

(-2,-3)

.

Check: The reflection gives

(-3,2)

. Rotating that point counterclockwise by a quarter-turn gives

(-2,-3)

.

Worked example

Dilate a triangle

Triangle

RST

has vertices

R(1,1)

,

S(3,1)

, and

T(2,3)

. Dilate it by a scale factor of 2, using the origin as the centre. Find the image vertices.

Triangle before and after dilation

Connect each set of vertices. The image is enlarged from the origin by scale factor 2.

Use the scale factor
The centre is the origin and the scale factor is 2. Multiply both coordinates of each point by 2.
$(x,y)\rightarrow(2x,2y)$
Calculate each image point
For $R(1,1)$ , double both coordinates to get $(2,2)$ . For $S(3,1)$ , get $(6,2)$ . For $T(2,3)$ , get $(4,6)$ .
$R'(2,2),\ S'(6,2),\ T'(4,6)$
Check the size change
Each image point is twice as far from the origin as its matching original point. Plot and join the image points in order.

Answer: The image vertices are

R'(2,2)

,

S'(6,2)

, and

T'(4,6)

.

Check: Every coordinate has been multiplied by 2, as required for a dilation centred at the origin with scale factor 2.

Common mistakes and how to avoid them

Changing the vertical coordinate when moving right.

Correction: A horizontal move changes the first coordinate,

x

. A vertical move changes the second coordinate,

y

.

Reflecting across the

x

-axis by changing the

x

-coordinate.

Correction: For a reflection across the

x

-axis, keep

x

and change the sign of

y

.

Rotating or reflecting around the wrong centre or line.

Correction: Identify the centre or line in the instruction before moving any point.

Dilating by multiplying only one coordinate.

Correction: For a dilation centred at the origin, multiply both coordinates by the scale factor.

Lesson summary

A translation slides every point by the same amount and in the same direction.
A reflection flips a figure across a line.
A rotation turns a figure around a centre by a stated amount and direction.
A dilation enlarges or reduces a figure from a centre using a scale factor.
Plot and label every image point, then join corresponding vertices in order.

Check your understanding

Question 1

Point

Q(2,-1)

is translated 3 units left and 2 units up. What are its image coordinates?

$(-1,1)$
$(5,1)$
$(-1,-3)$
$(1,2)$

Show answer and explanation

(-1,1)

Left 3 changes

x

from 2 to

-1

. Up 2 changes

y

from

-1

to 1.

Question 2

What is the reflection of

M(-4,2)

across the

x

-axis?

$(4,2)$
$(-4,-2)$
$(4,-2)$
$(-4,2)$

Show answer and explanation

(-4,-2)

A reflection across the

x

-axis keeps

x

and changes the sign of

y

.

Question 3

Point

N(2,-3)

is rotated

90^\circ

counterclockwise around the origin. Where does it move?

$(-3,-2)$
$(3,2)$
$(3,-2)$
$(-2,3)$

Show answer and explanation

(3,2)

A counterclockwise quarter-turn sends

(x,y)

to

(-y,x)

. For

(2,-3)

, this gives

(3,2)

. So the correct choice is

(3,2)

.

Key terms

Cartesian plane: A grid with a horizontal $x$ -axis and a vertical $y$ -axis.
Ordered pair: Two coordinates written in order as $(x,y)$ to name a point.
Image: The point or figure after a transformation.
Translation: A slide that moves every point the same distance and direction.
Reflection: A flip across a line.
Rotation: A turn around a fixed centre.
Dilation: An enlargement or reduction from a centre by a scale factor.
Scale factor: The number that tells how distances from the centre are multiplied in a dilation.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 8 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, rotate, and dilate figures on a Cartesian plane

Ontario Grade 8 Mathematics

Translations, reflections, rotations, and dilations

What you will learn

Read points and plan a move

Reflect and rotate a figure

Dilate a figure and check the result

Translate a triangle

Reflect, then rotate a point

Dilate a triangle

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through Grade 8 Math

About this lesson and its review