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

Reflecting, translating and rotating points and triangles on the Cartesian plane.

Gr 8 Term 4 — Transformation GeometryTerm 4

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What gets asked

Reflect a point or triangle across the x-axis or the y-axis.
Translate a point or triangle a given number of units.
Rotate a triangle about the origin.
Describe the transformation that maps a figure onto its image.

You must be able to

Change the correct coordinate when reflecting in the x-axis or y-axis.
Add or subtract units correctly when translating a point.
Keep a figure the same size and shape under a reflection or rotation.
Name a transformation fully, including direction and number of units.

Traps that cost marks

Changing the sign of the wrong coordinate when reflecting in an axis. Reflecting in the $x$ -axis changes the sign of $y$ . Reflecting in the $y$ -axis changes the sign of $x$ .
Adding an up-or-down shift to the $x$ -coordinate instead of the $y$ -coordinate. Left and right change the $x$ -value. Up and down change the $y$ -value.
Subtracting corresponding lengths to find an enlargement factor, instead of dividing them. The enlargement factor is the new length divided by the old length, not the difference.
Naming the type of transformation but leaving out the axis, direction or number of units. A full description needs the transformation type and its details, such as axis or units.

Worked example

Reflect the point

A (3; - 5)

in the

x

-axis. Give the coordinates of the image.

Reflecting in the $x$ -axis keeps the $x$ -value the same.
The $y$ -value changes sign: $- 5$ becomes $5$ .
The image is $A^{'} (3; 5)$ .

Reflect, translate and rotate

A shape can slide, flip or turn, and the numbers tell you exactly where it lands.

Cartesian plane: A grid made by two number lines crossing at right angles. The $x$ -axis runs across, the $y$ -axis up and down.
coordinates: The pair of numbers that fixes a point, written $(x; y)$ . The across value comes first.
origin: The point where the two axes cross. Its coordinates are $(0; 0)$ .
transformation: A change in the position of a figure. Flipping, sliding and turning are all transformations.
image: The figure you end up with after a transformation. The one you started with is the object.

Start at the origin of the Cartesian plane. The first number says how far across: right positive, left negative. The second says how far up or down: up positive, down negative. Order matters, so $(3; 5)$ and $(5; 3)$ differ.

A reflection is a flip over an axis, like a mirror. Reflecting in the $x$ -axis keeps the $x$ -value and changes the sign of the $y$ -value. Reflecting in the $y$ -axis does the opposite. The image stays the same size and shape.

A translation is a slide. Nothing turns and nothing changes size. Left and right change only the $x$ -value. Up and down change only the $y$ -value. Sliding $4$ down takes $4$ off each.

A rotation is a turn about a fixed point, usually the origin. Always say which way: clockwise or anticlockwise. A half turn is $18 0^{\circ}$ and changes the sign of both values. It lands in the same place either way, so direction matters most for a quarter turn.

Describing a transformation takes more than its name. A reflection needs its axis. A translation needs direction and units. A rotation needs centre, direction and angle.

Rules to remember

in the $x$ -axis: $(x; y)$ becomes $(x; - y)$
in the $y$ -axis: $(x; y)$ becomes $(- x; y)$

Examples

Reflect the point

P (- 4; 2)

in the

y

-axis. Give the image.

Worked answer

Reflecting in the $y$ -axis keeps the $y$ -value.
The $x$ -value changes sign, so $- 4$ becomes $4$ .
The image is $P^{'} (4; 2)$ .

Answer: $P^{'} (4; 2)$

Only one value changed, and the axis decided which one.

Triangle

A (1; 2)

B (4; 2)

C (1; 6)

slides

3

units left and

5

down. Give the image.

Worked answer

Left $3$ takes $3$ off every $x$ -value. Down $5$ takes $5$ off every $y$ -value.
$A (1; 2)$ becomes $A^{'} (- 2; - 3)$ .
$B (4; 2)$ becomes $B^{'} (1; - 3)$ .
$C (1; 6)$ becomes $C^{'} (- 2; 1)$ .

Answer: $A^{'} (- 2; - 3)$ , $B^{'} (1; - 3)$ , $C^{'} (- 2; 1)$

Every vertex moved the same amount, so the shape did not change.

Triangle

D (2; 1)

E (5; 1)

F (2; 3)

is given a half turn about the origin. Give the image and describe the turn in full.

Worked answer

A half turn about the origin changes the sign of both values.
$D (2; 1)$ becomes $D^{'} (- 2; - 1)$ , and $E (5; 1)$ becomes $E^{'} (- 5; - 1)$ .
$F (2; 3)$ becomes $F^{'} (- 2; - 3)$ .
In full: a rotation of $18 0^{\circ}$ about the origin.

Answer: $D^{'} (- 2; - 1)$ , $E^{'} (- 5; - 1)$ , $F^{'} (- 2; - 3)$

Calling it a rotation is not enough. The centre and angle finish the job.

Traps

Writing the two coordinates the wrong way round. Across first, then up or down. $(3; 5)$ and $(5; 3)$ are different points.
Changing the sign of the wrong coordinate when reflecting in an axis. Reflecting in the $x$ -axis changes the sign of $y$ . Reflecting in the $y$ -axis changes the sign of $x$ .
Adding an up-or-down shift to the $x$ -coordinate instead of the $y$ -coordinate. Left and right change the $x$ -value. Up and down change the $y$ -value.
Naming the type of transformation but leaving out the axis, direction or number of units. A full description needs the transformation type and its details, such as axis or units.

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

Enlarge and reduce

Blow a picture up or shrink it down: every length changes, and not one angle does.

enlargement: A transformation that makes a figure bigger. Every length is multiplied by the same number.
reduction: A transformation that makes a figure smaller. Every length is divided by the same number.
scale factor: The number every length is multiplied by. It is a new length divided by the matching old length.
corresponding sides: Sides that sit in the same place in the two figures. The longest matches the longest.

An enlargement or a reduction changes the size of a figure, not its shape. Every length changes by the same scale factor, and every angle stays exactly as it was. The image is a bigger or smaller copy. A slide, a flip or a turn is different: there nothing changes size.

To find the factor, divide. Never subtract. Take a length on the new figure and divide it by the corresponding side on the old one. A factor above $1$ means an enlargement. A factor below $1$ means a reduction.

Perimeter and area do not behave the same way. Perimeter is a length, so it is multiplied by the factor once. Area covers two directions, so the factor is used twice. Double every side and the area becomes $4$ times as big, not $2$ .

Rules to remember

factor $k$ : every length $\times k$
factor $k$ : perimeter $\times k$ , area $\times k^{2}$

Examples

A photo is

6

cm wide. It is enlarged until it is

18

cm wide. Find the scale factor.

Worked answer

Divide the new width by the old width.
$18 \div 6 = 3$ .
The factor is $3$ , which is above $1$ , so this is an enlargement.

Answer: $3$

Subtracting would give $12$ , and that is a difference, not a factor.

A rectangle is

5

cm by

3

cm. It is enlarged by a factor of

2

. Find the new perimeter and the new area.

Worked answer

New sides: $5 \times 2 = 10$ and $3 \times 2 = 6$ .
New perimeter: $2 \times (10 + 6) = 32$ .
The old perimeter was $2 \times (5 + 3) = 16$ , so it doubled.
New area: $10 \times 6 = 60$ .
The old area was $5 \times 3 = 15$ , and $15 \times 4 = 60$ .

Answer: perimeter $32$ cm, area $60$ cm $^{2}$

The perimeter doubled, but the area went up four times.

A poster

60

cm tall is reduced to

15

cm tall for a flyer. Find the factor, and say what happens to the area.

Worked answer

$15 \div 60 = \frac{1}{4}$ .
That is below $1$ , so this is a reduction.
The area uses the factor twice: $\frac{1}{4} \times \frac{1}{4} = \frac{1}{16}$ .
So the flyer covers a sixteenth of what the poster covered.

Answer: a factor of $\frac{1}{4}$ , and $\frac{1}{16}$ of the area

The factor was used twice again, so the area shrank far faster than the height.

Traps

Subtracting corresponding lengths to find an enlargement factor, instead of dividing them. The enlargement factor is the new length divided by the old length, not the difference.
Comparing a side with one that is not its match in the other figure. Match corresponding sides: longest with longest, shortest with shortest.
Doubling the area when every side length has been doubled. The length and the width both double, so the area is $2 \times 2 = 4$ times as big.
Treating the perimeter and the area in the same way. Perimeter is multiplied by the factor once. Area is multiplied by it twice.

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

Grade 8 Transformation Geometry

What gets asked

You must be able to

Traps that cost marks

Worked example

Reflect, translate and rotate

Rules to remember

Examples

Traps

Enlarge and reduce

Rules to remember

Examples

Traps

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