Transformation Geometry (Grade 9) — Mathematics Grades 8–11

Reflecting, translating and enlarging a point or figure on the Cartesian plane.

Gr 9 Term 4 — Transformation GeometryTerm 4

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What gets asked

Reflect a point in the x-axis, the y-axis or the line y = x.
Translate a point and give the co-ordinates of the image.
Find the co-ordinates of a figure after an enlargement or reduction.
Determine the scale factor of an enlargement or reduction.

You must be able to

Change the sign of the correct co-ordinate when reflecting in an axis.
Add or subtract units from each co-ordinate for a translation.
Multiply each co-ordinate by the scale factor for an enlargement.
Divide an image length by the matching original length to find the scale factor.

Traps that cost marks

Changing the y-value instead of the x-value when reflecting in the y-axis. Reflecting in the y-axis changes the sign of x only: (x ; y) becomes (-x ; y).
Changing both co-ordinates for a translation that only moves sideways or only up and down. A translation that moves only left or right changes x only. One that moves only up or down changes y only.
Adding the scale factor to each co-ordinate instead of multiplying. An enlargement multiplies every co-ordinate by the scale factor.

Worked example

A(3 ; -2) is reflected in the x-axis to give A′. Write down the co-ordinates of A′.

Reflecting in the x-axis changes the sign of the y-value only.
The x-value, 3, stays the same.
A′ = (3 ; 2).

Reflect and translate

Flipping and sliding a point on the grid, and reading the answer straight off its two numbers.

co-ordinates: The pair of numbers that fixes a point: how far across, then how far up.
transformation: A move that changes where a figure sits: a flip, a slide or a turn.
image: The point or figure you end up with after a transformation.
reflection: A flip over a line. The image sits the same distance from that line, on the other side.
translation: A slide. Every point moves the same distance the same way, and nothing turns.
quadrant: One of the four parts that the two axes cut the grid into.

A point is written $(x; y)$ . The first number says how far across, the second how far up. Swapping the two co-ordinates by mistake puts the point somewhere else entirely.

Reflecting in the $x$ -axis flips the point over the flat axis. The across value stays as it is and the up value changes sign, so $(x; y)$ becomes $(x; - y)$ .

Reflecting in the $y$ -axis flips it over the upright axis instead. Now the across value is the one that changes sign: $(x; y)$ becomes $(- x; y)$ . The axis you reflect in is the one that stays put.

Reflecting in the line $y = x$ swaps the two numbers over, so $(x; y)$ becomes $(y; x)$ . Nothing changes sign here.

A translation slides the figure. Right adds to the across value and left subtracts from it. Up adds to the up value and down subtracts from it. A slide one way only changes one number.

Rules to remember

in the $x$ -axis: $(x; y) \to (x; - y)$
in the $y$ -axis: $(x; y) \to (- x; y)$
in the line $y = x$ : $(x; y) \to (y; x)$

Examples

P (- 5; 2)

is reflected in the

y

-axis. Give the co-ordinates of the image

P^{'}

Worked answer

Reflecting in the $y$ -axis changes the sign of the across value only.
The across value is $- 5$ , and $- (- 5) = 5$ .
The up value, $2$ , does not move.
So $P^{'}$ is $(5; 2)$ .

Answer: $(5; 2)$

The axis you reflect in stays where it is, so the up value was left alone.

B (- 2; 5)

is translated

6

units right and

3

units down. Give the co-ordinates of

B^{'}

Worked answer

Right adds to the across value: $- 2 + 6 = 4$ .
Down subtracts from the up value: $5 - 3 = 2$ .
Now put the two new numbers together.
So $B^{'}$ is $(4; 2)$ .

Answer: $(4; 2)$

The point slid out of the second quadrant into the first, and the signs looked after themselves.

C (7; 1)

has image

C^{'} (1; 7)

. Name the transformation.

Worked answer

The two numbers have swapped over.
No sign has changed, so it is not a reflection in an axis.
Nothing was added on, so it is not a translation.
Swapping the numbers is a reflection in the line $y = x$ .

Answer: a reflection in the line $y = x$

Ask one question: were the numbers swapped, sign-changed, or added to?

Traps

Changing the y-value instead of the x-value when reflecting in the y-axis. Reflecting in the y-axis changes the sign of x only: (x ; y) becomes (-x ; y).
Changing both co-ordinates for a translation that only moves sideways or only up and down. A translation that moves only left or right changes x only. One that moves only up or down changes y only.
Swapping the two co-ordinates AND changing their signs for the line $y = x$ . That reflection only swaps them. $(4; 9)$ becomes $(9; 4)$ , both still positive.
Calling it a translation when the two co-ordinates were swapped over. Swapped co-ordinates mean a reflection in the line $y = x$ , not a slide.

Learn and practise “Reflect and translate” in the app →

Enlarge and reduce

Growing or shrinking a figure by one number, and what that does to the distance round it and the space inside.

enlargement: A transformation that makes a figure bigger, with every length multiplied by the same number.
reduction: The same move the other way, making the figure smaller while keeping its shape.
scale factor: The number every length is multiplied by. Above $1$ it enlarges, below $1$ it reduces.
vertex: A corner of a figure. Two or more of them are called vertices.

An enlargement multiplies. Take each vertex and multiply both of its co-ordinates by the scale factor. Adding the factor on instead just slides the figure and leaves its size alone.

Multiply BOTH numbers of a vertex, never only one. Scaling the across value and leaving the up value squashes the shape rather than growing it.

A scale factor above $1$ gives an enlargement. A factor between $0$ and $1$ gives a reduction. A factor of exactly $1$ leaves the figure as it was.

To find the scale factor, divide an image length by the matching original length. Image over original, in that order. Turning the division round gives you the factor for going back the other way.

Doubling every length doubles the perimeter, because perimeter is just lengths added up. The area does NOT double: it becomes four times as big, since both the length and the width grew. The angles never change.

Rules to remember

enlarge by $k$ : $(x; y) \to (k x; k y)$
scale factor $= \frac{image length}{original length}$
lengths times $k$ means area times $k^{2}$

Examples

Triangle

A B C

has vertices

A (1; 2)

B (3; 2)

and

C (3; 5)

. Enlarge it about the origin by a scale factor of

2

Worked answer

Multiply both co-ordinates of each vertex by $2$ .
$A (1; 2)$ gives $A^{'} (2; 4)$ .
$B (3; 2)$ gives $B^{'} (6; 4)$ .
$C (3; 5)$ gives $C^{'} (6; 10)$ .

Answer: $A^{'} (2; 4)$ , $B^{'} (6; 4)$ , $C^{'} (6; 10)$

Every number was multiplied, so the shape stayed the same and only the size grew.

A photo

10

cm wide is enlarged until it is

25

cm wide. Find the scale factor.

Worked answer

Scale factor is the image length divided by the original length.
So it is $25 \div 10$ .
$25 \div 10 = 2, 5$ .
The scale factor is $2, 5$ .

Answer: $2, 5$

Dividing the other way gives $0, 4$ , which would shrink the photo instead.

A vegetable bed is

4

m by

3

m. Every side is doubled. What happens to its perimeter and its area?

Worked answer

Perimeter now: $2 \times (4 + 3) = 14$ m.
Area now: $4 \times 3 = 12$ square metres.
Doubled, the bed is $8$ m by $6$ m, so the perimeter is $2 \times (8 + 6) = 28$ m.
The new area is $8 \times 6 = 48$ square metres.

Answer: the perimeter doubles; the area is four times bigger

Lengths were multiplied by $2$ , so the area was multiplied by $2 \times 2 = 4$ .

Traps

Adding the scale factor to each co-ordinate instead of multiplying. An enlargement multiplies every co-ordinate by the scale factor.
Multiplying only one co-ordinate of each vertex. Multiply BOTH numbers of every vertex, or the shape comes out squashed.
Dividing the original by the image, which turns the factor upside down. Scale factor is image length divided by original length, in that order.
Multiplying the area by the scale factor instead of by its square. Lengths use $k$ , area uses $k \times k$ . A factor of $3$ makes the area $9$ times bigger.

Learn and practise “Enlarge and reduce” in the app →

Grade 9 Transformation Geometry

What gets asked

You must be able to

Traps that cost marks

Worked example

Reflect and translate

Rules to remember

Examples

Traps

Enlarge and reduce

Rules to remember

Examples

Traps

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