These positions illustrate several important patterns:

• (3, 2) is right and up.
• (−3, 2) is left and up.
• (−3, −2) is left and down.
• (3, −2) is right and down.
• (0, 4) lies on the y-axis.
• (−4, 0) lies on the x-axis.

---

Positive Coordinates

Consider:

P(5, 3)

Both coordinates are positive.

Start at:

(0, 0)

Move:

5 units right

and:

3 units up

The point is located in the upper-right part of the coordinate plane.

---

Negative x-Coordinates

Consider:

Q(−4, 2)

The x-coordinate is negative.

Move:

4 units left

Then the positive y-coordinate tells us to move:

2 units up

---

Negative y-Coordinates

Consider:

R(3, −5)

Move:

3 units right

Then:

5 units down

because the y-coordinate is negative.

---

Two Negative Coordinates

Consider:

S(−2, −4)

Move:

2 units left

and:

4 units down

Both coordinates are negative.

---

The Four Quadrants

The x-axis and y-axis divide the coordinate plane into four sections called quadrants.

The quadrants are numbered using Roman numerals:

Quadrant I

Quadrant II

Quadrant III

Quadrant IV

They are numbered counterclockwise beginning in the upper-right region.

---

Quadrant I

In Quadrant I:

x is positive

y is positive

Sign pattern:

(+, +)

Example:

(4, 6)

---

Quadrant II

In Quadrant II:

x is negative

y is positive

Sign pattern:

(−, +)

Example:

(−3, 7)

---

Quadrant III

In Quadrant III:

x is negative

y is negative

Sign pattern:

(−, −)

Example:

(−5, −2)

---

Quadrant IV

In Quadrant IV:

x is positive

y is negative

Sign pattern:

(+, −)

Example:

(6, −4)

---

Remembering the Quadrants

The quadrants begin in the upper-right and move counterclockwise.

---

Points on the x-Axis

A point on the x-axis has:

y = 0

Examples:

(5, 0)

(−3, 0)

(10, 0)

These points are not located inside any quadrant.

For example:

(−6, 0)

means move 6 units left from the origin but do not move up or down.

---

Points on the y-Axis

A point on the y-axis has:

x = 0

Examples:

(0, 5)

(0, −7)

(0, 12)

These points are also not inside any quadrant.

---

The Origin Is Special

The origin:

(0, 0)

lies on both:

• The x-axis.
• The y-axis.

It does not belong to any quadrant.

---

Reading Coordinates from a Graph

Sometimes a point is already plotted and you must determine its coordinates.

To read a point:

Step 1: Determine its horizontal position.

This gives the:

x-coordinate

Step 2: Determine its vertical position.

This gives the:

y-coordinate

Step 3: Write the coordinates as:

(x, y)

---

Example: Reading a Point

Suppose point A is located:

• 4 units to the right of the origin.
• 3 units above the origin.

Then:

x = 4

y = 3

Therefore:

A(4, 3)

---

Example with Negative Coordinates

Suppose point B is located:

• 5 units left.
• 2 units down.

Then:

x = −5

y = −2

Therefore:

B(−5, −2)

Point B lies in:

Quadrant III

---

Coordinates Are Ordered

The points:

(2, 5)

and:

(5, 2)

are not the same point.

For:

(2, 5)

move:

2 right, 5 up

For:

(5, 2)

move:

5 right, 2 up

Changing the order changes the location.

---

Comparing Coordinates

Coordinates can help us compare the positions of points.

Suppose:

A(2, 5)

B(2, −3)

Both points have:

x = 2

Therefore, they are vertically aligned.

Point A is above point B because:

5 > −3

---

Same y-Coordinate

Suppose:

C(−4, 3)

D(6, 3)

Both have:

y = 3

Therefore, the points are horizontally aligned.

Point C is to the left of D because:

−4 < 6

---

Vertical and Horizontal Lines

If several points have the same x-coordinate, they form a:

vertical line

For example:

(3, 5)

(3, 2)

(3, −4)

all have:

x = 3

---

If several points have the same y-coordinate, they form a:

horizontal line

For example:

(−4, 2)

(0, 2)

(6, 2)

all have:

y = 2

---

Distance Along a Horizontal Line

Coordinates can be used to determine distances.

Suppose:

A(2, 4)

and:

B(8, 4)

The y-coordinates are equal, so the points lie on a horizontal line.

The distance is:

8 − 2 = 6 units

---

Distance Along a Vertical Line

Suppose:

C(−3, 2)

and:

D(−3, 9)

The x-coordinates are equal.

Distance:

9 − 2 = 7 units

---

Distance Across Zero

Be careful when coordinates are on opposite sides of zero.

Suppose:

A(−4, 2)

and:

B(3, 2)

The horizontal distance is:

3 − (−4)

= 7 units

You can also think:

4 units to reach zero + 3 more units

= 7 units

---

Plotting Shapes

Coordinate planes can be used to create geometric shapes.

Consider the points:

A(1, 1)

B(5, 1)

C(5, 4)

D(1, 4)

Connecting the points in order creates a:

rectangle

The coordinates tell us exactly where each vertex is located.

---

Finding Missing Coordinates

Suppose three vertices of a rectangle are:

A(2, 1)

B(7, 1)

C(7, 5)

The fourth vertex must align:

• Horizontally with C.
• Vertically with A.

Therefore:

D(2, 5)

Coordinates can therefore be used to reason about geometric shapes.

---

Reflections Across the x-Axis

A reflection creates a mirror image.

When a point is reflected across the x-axis:

the x-coordinate stays the same

and:

the sign of the y-coordinate changes

For example:

(4, 3) → (4, −3)

Another example:

(−2, −5) → (−2, 5)

---

Reflections Across the y-Axis

When a point is reflected across the y-axis:

the y-coordinate stays the same

and:

the sign of the x-coordinate changes

For example:

(4, 3) → (−4, 3)

Another example:

(−6, 2) → (6, 2)

---

Reflection Through the Origin

When a point is reflected through the origin, both signs change.

For example:

(3, 5) → (−3, −5)

and:

(−4, 2) → (4, −2)

---

Coordinate Changes and Movement

Coordinates can also describe movement.

Suppose a point moves from:

(2, 3)

to:

(7, 3)

The x-coordinate increases by:

5

The y-coordinate does not change.

Therefore, the point moved:

5 units right

---

Another Movement Example

A point moves from:

(4, 6)

to:

(4, 1)

The x-coordinate remains the same.

The y-coordinate decreases by:

5

Therefore, the point moved:

5 units down

---

Using Coordinates to Describe Locations

Coordinate systems allow us to describe positions precisely.

For example, imagine a grid map where:

School = (2, 4)

Park = (−3, 5)

Library = (4, −2)

Sports field = (−2, −3)

Instead of giving long descriptions, each location can be identified using an ordered pair.

---

Coordinates in Maps

Maps often use grid systems similar to coordinate planes.

Coordinates can help identify:

• Locations.
• Distances.
• Directions.
• Routes.

Modern geographic systems are more complicated than a basic classroom coordinate grid, but the underlying idea is similar:

numbers are used to describe precise locations.

---

Coordinates in Computer Graphics

Computer screens also use coordinate systems.

Images, shapes, buttons, and other objects can be positioned using coordinates.

Coordinates are important in:

• Computer graphics.
• Animation.
• Video games.
• Digital design.
• Robotics.

A computer needs precise numerical information about where objects should appear.

---

Coordinates in Science

Coordinate systems are extremely important in science.

In physics, graphs may represent quantities such as:

• Distance and time.
• Velocity and time.
• Force and displacement.
• Temperature and time.

The horizontal axis usually represents one variable, while the vertical axis represents another.

Understanding coordinate planes therefore prepares learners for graphing scientific data.

---

Coordinates in Mathematics

Coordinate planes connect geometry and algebra.

Geometry describes:

• Shapes.
• Locations.
• Distances.

Algebra describes relationships using:

• Numbers.
• Variables.
• Equations.

Coordinates allow shapes to be described using numbers and equations.

This branch of mathematics is called:

coordinate geometry

or:

analytic geometry

---

Scale on a Coordinate Plane

Not every coordinate grid counts by ones.

A graph might increase by:

1

2

5

10

or another amount.

For example, if each grid interval represents:

5 units

then the first mark after zero represents:

5

the second represents:

10

and the third represents:

15

Always check the scale before reading coordinates.

---

Example with a Different Scale

Suppose each grid square represents:

10 units

A point located:

• 3 squares right.
• 2 squares up.

has coordinates:

(30, 20)

not:

(3, 2)

---

A Reliable Method for Plotting Points

When plotting an ordered pair:

(x, y)

use this method:

Step 1: Start at the origin.

Step 2: Read the x-coordinate.

Move:

• Right if positive.
• Left if negative.

Step 3: Read the y-coordinate.

Move:

• Up if positive.
• Down if negative.

Step 4: Mark the point.

Step 5: Label it if required.

A useful memory aid is:

x first, y second

or:

across, then up or down

---

A Reliable Method for Reading Points

When determining the coordinates of an existing point:

Step 1: Trace vertically to the x-axis.

Read the:

x-coordinate

Step 2: Trace horizontally to the y-axis.

Read the:

y-coordinate

Step 3: Write:

(x, y)

Step 4: Check whether the signs match the point's quadrant.

---

Common Mistakes

Reversing x and y

For:

(3, 7)

3 is the x-coordinate.

7 is the y-coordinate.

Do not write:

(7, 3)

---

Moving Vertically First

The first coordinate describes horizontal movement.

Always think:

x first → horizontal

y second → vertical

---

Forgetting Negative Signs

A point to the left of the y-axis has a:

negative x-coordinate

A point below the x-axis has a:

negative y-coordinate

---

Numbering Quadrants Clockwise

Quadrants are numbered:

counterclockwise

starting in the upper-right.

---

Saying Points on Axes Are in Quadrants

A point such as:

(5, 0)

is on the x-axis.

It is not in Quadrant I or IV.

Similarly:

(0, −4)

is on the y-axis.

It is not in Quadrant III or IV.

---

Forgetting the Scale

Do not assume every grid square represents one unit.

Check the numbers printed on the axes.

---

Confusing the Origin with an Axis

The origin is one specific point:

(0, 0)

The axes are entire number lines.

---

Check Your Understanding

1. What is a coordinate plane?

2. Which axis is horizontal?

3. Which axis is vertical?

4. What is the origin?

5. Write the coordinates of the origin.

6. In the ordered pair (4, 7), which number is the x-coordinate?

7. In the ordered pair (4, 7), which number is the y-coordinate?

8. Describe how you would plot (5, 3).

9. Describe how you would plot (−4, 6).

10. Describe how you would plot (3, −5).

11. Describe how you would plot (−2, −7).

12. In which quadrant is (6, 4)?

13. In which quadrant is (−3, 8)?

14. In which quadrant is (−5, −2)?

15. In which quadrant is (7, −4)?

16. Does the point (5, 0) lie in a quadrant? Explain.

17. Where is the point (0, −6) located?

18. What must be true about the y-coordinate of every point on the x-axis?

19. What must be true about the x-coordinate of every point on the y-axis?

20. Explain why (2, 5) and (5, 2) represent different locations.

21. Points A(3, 7) and B(3, 1) have the same x-coordinate. How are they aligned?

22. Points C(−4, 5) and D(6, 5) have the same y-coordinate. How are they aligned?

23. Find the horizontal distance between (2, 4) and (9, 4).

24. Find the vertical distance between (−3, −2) and (−3, 6).

25. Find the horizontal distance between (−5, 3) and (4, 3).

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

27. Reflect (−3, 5) across the y-axis.

28. A point moves from (2, 4) to (8, 4). Describe its movement.

29. A point moves from (−3, 6) to (−3, 1). Describe its movement.

30. Give two real-world situations in which coordinates can be used to describe locations or positions.

---

Key Terms

• Coordinate plane – two-dimensional grid formed by perpendicular number lines.
• x-axis – horizontal axis of a coordinate plane.
• y-axis – vertical axis of a coordinate plane.
• Origin – point where the axes intersect; (0, 0).
• Ordered pair – pair of numbers written (x, y) describing a point's location.
• x-coordinate – first number in an ordered pair; describes horizontal position.
• y-coordinate – second number in an ordered pair; describes vertical position.
• Quadrant – one of the four regions formed by the coordinate axes.
• Plot – place a point at its correct location on a coordinate plane.
• Coordinate – number used to describe a point's position.
• Scale – amount represented by each interval on an axis.
• Reflection – transformation that creates a mirror image.
• Coordinate geometry – study of geometric figures using coordinates and algebra.

---

Key Takeaways

• A coordinate plane contains a horizontal x-axis and vertical y-axis.
• The axes intersect at the origin.
• The origin is:

(0, 0)

• Locations are written as ordered pairs:

(x, y)

• The x-coordinate always comes first.
• Positive x means move right.
• Negative x means move left.
• Positive y means move up.
• Negative y means move down.
• Quadrant I has signs (+, +).
• Quadrant II has signs (−, +).
• Quadrant III has signs (−, −).
• Quadrant IV has signs (+, −).
• Points on the x-axis have y = 0.
• Points on the y-axis have x = 0.
• Points on either axis are not in a quadrant.
• Coordinates can describe locations, movements, shapes, and distances.
• Coordinate systems are widely used in mathematics, science, maps, engineering, computing, and navigation.

A useful rule for plotting any point is:

Start at the origin → x first → y second.

Geometric Transformations

A geometric transformation changes the position, orientation, or size of a figure.

Two important transformations are:

• Reflection – flips a figure across a line.
• Rotation – turns a figure around a fixed point.

Like translations, reflections and rotations are rigid transformations. This means they preserve:

• Side lengths.
• Angle measures.
• Size.
• Shape.

The original figure and its transformed image are therefore congruent.

---

Reflections

A reflection is a transformation that flips a figure across a line.

The result looks like a mirror image of the original figure.

The line across which the figure is flipped is called the:

line of reflection

Imagine placing a mirror along this line. The reflected figure would appear on the opposite side.

---

The Line of Reflection

Every point and its reflected image are the same perpendicular distance from the line of reflection.

For example, if point A is:

3 units to the right

of a vertical line of reflection, then A′ will be:

3 units to the left

of that line.

The line of reflection lies exactly halfway between corresponding points.

---

Reflection Across the x-Axis

One of the most common coordinate transformations is reflection across the x-axis.

Suppose:

A(3, 4)

is reflected across the x-axis.

The x-coordinate stays the same.

The y-coordinate changes sign.

Therefore:

A(3, 4) → A′(3, −4)

The rule is:

(x, y) → (x, −y)

---

Why the y-Coordinate Changes

The x-axis is horizontal.

A reflection across it changes the point's vertical position.

For example:

(2, 5) → (2, −5)

The original point is 5 units above the x-axis.

The reflected point must be 5 units below the x-axis.

---

Another x-Axis Example

Reflect:

B(−4, −3)

across the x-axis.

Keep x:

−4

Change the sign of y:

−3 → 3

Therefore:

B′(−4, 3)

---

Reflection Across the y-Axis

When reflecting across the y-axis, the y-coordinate stays the same.

The x-coordinate changes sign.

For example:

C(4, 2) → C′(−4, 2)

The rule is:

(x, y) → (−x, y)

---

Why the x-Coordinate Changes

The y-axis is vertical.

A reflection across it changes the point's horizontal position.

For example:

(6, 3)

is 6 units to the right of the y-axis.

Its reflection must be 6 units to the left.

Therefore:

(6, 3) → (−6, 3)

---

Reflection Across the Line y = x

Another important line of reflection is:

y = x

When reflecting across this line, the x- and y-coordinates exchange positions.

The rule is:

(x, y) → (y, x)

For example:

(2, 5) → (5, 2)

and:

(−3, 4) → (4, −3)

---

Reflecting an Entire Figure

Suppose triangle ABC has vertices:

A(1, 2)

B(4, 2)

C(2, 5)

Reflect the triangle across the y-axis.

Use:

(x, y) → (−x, y)

Therefore:

A(1, 2) → A′(−1, 2)

B(4, 2) → B′(−4, 2)

C(2, 5) → C′(−2, 5)

Every vertex follows the same reflection rule.

---

What Reflections Preserve

A reflection preserves:

• Side lengths.
• Angle measures.
• Parallel lines.
• Perpendicular lines.
• Area.
• Perimeter.
• Overall size and shape.

However, a reflection reverses orientation.

This is an important difference between reflections and translations.

---

Orientation After Reflection

Imagine a triangle whose vertices go around the figure in this order:

A → B → C

After reflection, its mirror image has the opposite orientation.

This is similar to looking at writing in a mirror.

The shape has not changed size, but it has been:

flipped

---

Lines of Reflection in Shapes

A line of reflection can also refer to a line of symmetry within a figure.

If a shape can be folded along a line so that the two halves match exactly, that line is a:

line of symmetry

or:

line of reflection

---

Lines of Reflection in a Square

A square has:

4 lines of reflection

These are:

• One vertical line through the center.
• One horizontal line through the center.
• Two diagonals.

Each line divides the square into two matching halves.

---

Lines of Reflection in a Rectangle

A non-square rectangle has:

2 lines of reflection

They pass:

• Vertically through the center.
• Horizontally through the center.

Its diagonals are not lines of reflection unless the rectangle is a square.

---

Lines of Reflection in Triangles

An equilateral triangle has:

3 lines of reflection

An isosceles triangle has:

1 line of reflection

A scalene triangle has:

0 lines of reflection

This provides another way to compare triangle types.

---

Lines of Reflection in Circles

A circle has:

infinitely many lines of reflection

Any line passing through its center divides the circle into two matching halves.

---

Rotations

A rotation is a transformation that turns a figure around a fixed point.

The fixed point is called the:

center of rotation

A rotation must specify:

• Center of rotation.
• Angle of rotation.
• Direction of rotation.

---

Clockwise and Counterclockwise

A rotation can occur in two directions.

Clockwise means turning in the same direction as the hands of a clock.

Counterclockwise means turning in the opposite direction.

For example:

90° clockwise

and:

90° counterclockwise

produce different positions.

---

Common Rotation Angles

Common rotation angles include:

90°

180°

270°

360°

A 360° rotation is one complete turn.

After a full 360° rotation, a figure returns to its original position.

---

Quarter, Half, and Full Turns

It can be helpful to connect angles with turns.

90° = ¼ turn

180° = ½ turn

270° = ¾ turn

360° = 1 complete turn

---

Rotations About the Origin

On a coordinate plane, rotations are often performed around:

the origin (0, 0)

There are useful coordinate rules for these rotations.

For a 90° counterclockwise rotation:

(x, y) → (−y, x)

For a 180° rotation:

(x, y) → (−x, −y)

For a 270° counterclockwise rotation:

(x, y) → (y, −x)

---

90° Counterclockwise Rotation

Consider:

A(3, 2)

Rotate it 90° counterclockwise around the origin.

Use:

(x, y) → (−y, x)

Therefore:

(3, 2) → (−2, 3)

---

Another 90° Counterclockwise Example

Consider:

B(−4, 2)

Apply:

(x, y) → (−y, x)

Therefore:

B(−4, 2) → B′(−2, −4)

The point moves from Quadrant II to Quadrant III.

---

90° Clockwise Rotation

For a 90° clockwise rotation around the origin:

(x, y) → (y, −x)

For example:

(3, 2) → (2, −3)

Notice that this is the same coordinate rule as a:

270° counterclockwise rotation

---

180° Rotation

For a 180° rotation around the origin:

(x, y) → (−x, −y)

Both coordinates change signs.

For example:

(4, 3) → (−4, −3)

Another example:

(−5, 2) → (5, −2)

The direction does not matter for a 180° rotation.

A 180° clockwise rotation and a 180° counterclockwise rotation produce the same result.

---

270° Counterclockwise Rotation

For a 270° counterclockwise rotation:

(x, y) → (y, −x)

For example:

(2, 5) → (5, −2)

A 270° counterclockwise rotation is equivalent to:

90° clockwise

---

Rotation Rules About the Origin

These rules apply when the center of rotation is the origin.

---

Understanding Rather Than Memorizing

Rotation rules can be memorized, but understanding the movement is more useful.

Consider:

A(4, 1)

The point lies mostly to the right and slightly above the origin.

After a 90° counterclockwise turn, it should lie:

• Mostly above the origin.
• Slightly to the left.

The result:

A′(−1, 4)

makes sense.

A quick sketch can help check whether a coordinate rule has been used correctly.

---

Rotating an Entire Figure

Suppose triangle ABC has:

A(1, 1)

B(4, 1)

C(2, 3)

Rotate it 90° counterclockwise around the origin.

Use:

(x, y) → (−y, x)

For A:

A(1, 1) → A′(−1, 1)

For B:

B(4, 1) → B′(−1, 4)

For C:

C(2, 3) → C′(−3, 2)

The entire triangle has turned around the origin.

---

Rotating About a Point Other Than the Origin

A figure can rotate around any point.

Suppose point A is rotated around point P.

Point P remains fixed while A travels along part of a circular path around it.

The distance between A and P remains constant.

This gives us an important rotation property:

Every point remains the same distance from the center of rotation.

---

Finding the Center of Rotation

When examining a diagram, the center of rotation is the fixed point around which the figure turns.

To identify it, consider corresponding points such as:

A and A′

The center must be equally distant from both.

The same must be true for:

B and B′

and other corresponding points.

For basic problems, the center is often given directly.

---

Rotations Preserve Size and Shape

Rotations are rigid transformations.

They preserve:

• Side lengths.
• Angles.
• Area.
• Perimeter.
• Parallel relationships.
• Perpendicular relationships.

Therefore, the original figure and rotated image are:

congruent

---

Rotations Preserve Orientation

Unlike a reflection, a rotation does not create a mirror image.

A rotation turns the figure but preserves the order of its vertices.

This means its orientation is preserved.

---

Comparing Reflections and Rotations

Reflections and rotations are both rigid transformations.

They both preserve:

• Size.
• Shape.
• Side lengths.
• Angle measures.
• Area.
• Perimeter.

However, they move figures differently.

---

Reflection Versus Rotation Example

Suppose an arrow points:

right

After reflection across a vertical line, it may point:

left

It has been flipped.

After a 90° counterclockwise rotation, it would point:

up

It has been turned.

The two transformations therefore produce different results even though both preserve the arrow's size and shape.

---

Comparing Translation, Reflection, and Rotation

The three rigid transformations can be remembered as:

Translation → slide

Reflection → flip

Rotation → turn

All three preserve:

• Size.
• Shape.
• Lengths.
• Angles.

But they change position and sometimes orientation.

---

Reflection Rules

Some useful coordinate rules are:

Across the x-axis

(x, y) → (x, −y)

Across the y-axis

(x, y) → (−x, y)

Across y = x

(x, y) → (y, x)

---

Rotation Rules

For rotations around the origin:

90° counterclockwise

(x, y) → (−y, x)

180°

(x, y) → (−x, −y)

270° counterclockwise

(x, y) → (y, −x)

---

Worked Reflection Example

Rectangle ABCD has vertices:

A(1, 2)

B(5, 2)

C(5, 4)

D(1, 4)

Reflect the rectangle across the y-axis.

Use:

(x, y) → (−x, y)

Therefore:

A′(−1, 2)

B′(−5, 2)

C′(−5, 4)

D′(−1, 4)

The new rectangle has the same dimensions as the original but has been flipped across the y-axis.

---

Worked Rotation Example

Triangle ABC has vertices:

A(2, 1)

B(5, 1)

C(3, 4)

Rotate the triangle:

180° about the origin

Use:

(x, y) → (−x, −y)

Therefore:

A′(−2, −1)

B′(−5, −1)

C′(−3, −4)

---

Checking a Reflection

Suppose:

A(4, 3) → A′(−4, 3)

The y-coordinate remained unchanged.

The x-coordinate changed sign.

Therefore, this is consistent with a reflection across the:

y-axis

---

Checking a Rotation

Suppose:

B(3, 5) → B′(−3, −5)

Both coordinates changed signs.

Therefore, this is consistent with:

a 180° rotation around the origin

---

Symmetry and Reflection

A figure has reflectional symmetry if it can be reflected across a line and match itself exactly.

Examples include:

• Squares.
• Rectangles.
• Equilateral triangles.
• Isosceles triangles.
• Circles.

The number of lines of reflection depends on the shape.

---

Rotational Symmetry

A figure has rotational symmetry if it can be rotated less than 360° around a point and still match its original position.

For example, a square matches itself after rotations of:

90°, 180°, 270°, and 360°

A non-square rectangle matches itself after:

180° and 360°

---

Order of Rotational Symmetry

The order of rotational symmetry tells us how many times a figure matches itself during one complete 360° turn.

A square has rotational symmetry of order:

4

A non-square rectangle has order:

2

An equilateral triangle has order:

3

A general scalene triangle has order:

1

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Reflections and Rotations in the Real World

These transformations appear frequently in nature, design, art, and engineering.

Examples include:

• Reflections in mirrors.
• Butterfly wing patterns.
• Building designs.
• Decorative tiles.
• Wheels and gears.
• Fans and propellers.
• Logos.
• Flowers.
• Snowflake patterns.

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A Reliable Reflection Strategy

When reflecting a figure:

Step 1: Identify the line of reflection.

Step 2: Determine each vertex's perpendicular distance from the line.

Step 3: Place the reflected point the same distance on the opposite side.

Step 4: Repeat for every vertex.

Step 5: Connect the new points in the corresponding order.

Step 6: Check that size and shape have been preserved.

---

A Reliable Rotation Strategy

When rotating a figure:

Step 1: Identify the center of rotation.

Step 2: Identify the angle.

Step 3: Identify whether the rotation is clockwise or counterclockwise.

Step 4: Rotate each vertex by the same angle.

Step 5: Plot the new coordinates.

Step 6: Connect the corresponding vertices.

Step 7: Check that every point remains the same distance from the center of rotation.

---

Common Mistakes

Confusing Reflection and Rotation

Remember:

Reflection = flip

Rotation = turn

---

Changing the Size of the Figure

Neither reflection nor rotation changes size.

The image must remain:

congruent

to the original.

---

Using the Wrong Reflection Rule

Across the x-axis:

change the sign of y

Across the y-axis:

change the sign of x

---

Changing Both Signs for Every Reflection

Changing both signs:

(x, y) → (−x, −y)

is not generally a reflection across the x- or y-axis.

It represents a:

180° rotation about the origin

---

Forgetting the Direction of Rotation

A 90° clockwise rotation is different from a 90° counterclockwise rotation.

Always check the direction.

---

Rotating Around the Wrong Point

A rotation must occur around a specific:

center of rotation

Rotating around the origin when another center is specified will produce the wrong result.

---

Forgetting That 180° Is Special

A 180° clockwise rotation produces the same result as a 180° counterclockwise rotation.

---

Assuming a Diagonal Is Always a Line of Symmetry

The diagonals of a square are lines of symmetry.

The diagonals of a non-square rectangle are not.

---

Confusing Reflectional and Rotational Symmetry

Reflectional symmetry involves a:

flip across a line

Rotational symmetry involves a:

turn around a point

A shape may have one, both, or neither.

---

Check Your Understanding

1. Define a reflection.

2. Define a rotation.

3. What is a line of reflection?

4. What is the center of rotation?

5. What three pieces of information describe a rotation?

6. Reflect (4, 3) across the x-axis.

7. Reflect (−5, 2) across the x-axis.

8. Reflect (6, −3) across the y-axis.

9. Reflect (−2, −7) across the y-axis.

10. Reflect (3, 5) across y = x.

11. Write the coordinate rule for reflection across the x-axis.

12. Write the coordinate rule for reflection across the y-axis.

13. Rotate (3, 2) 90° counterclockwise around the origin.

14. Rotate (4, 1) 90° clockwise around the origin.

15. Rotate (5, 3) 180° around the origin.

16. Rotate (−2, 4) 270° counterclockwise around the origin.

17. Write the rule for a 90° counterclockwise rotation around the origin.

18. Write the rule for a 180° rotation around the origin.

19. Explain why a reflection preserves size and shape.

20. Explain why a rotation preserves size and shape.

21. What happens to orientation during a reflection?

22. What happens to orientation during a rotation?

23. How many lines of reflection does a square have?

24. How many lines of reflection does a non-square rectangle have?

25. How many lines of reflection does an equilateral triangle have?

26. How many lines of reflection does a scalene triangle have?

27. What is the order of rotational symmetry of a square?

28. A point changes from (4, 6) to (−4, 6). Identify the likely transformation.

29. A point changes from (3, −5) to (−3, 5). Identify the rotation about the origin.

30. Explain one similarity and one difference between a reflection and a rotation.

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Key Terms

• Transformation – change in the position, orientation, or size of a geometric figure.
• Reflection – transformation that flips a figure across a line.
• Line of reflection – line across which a figure is reflected.
• Rotation – transformation that turns a figure around a fixed point.
• Center of rotation – fixed point around which a rotation occurs.
• Clockwise – direction in which the hands of a clock turn.
• Counterclockwise – direction opposite to clock hands.
• Preimage – original figure before a transformation.
• Image – resulting figure after a transformation.
• Congruent – having the same size and shape.
• Orientation – arrangement or direction of a figure and its vertices.
• Reflectional symmetry – symmetry created when a figure matches itself after reflection.
• Rotational symmetry – symmetry created when a figure matches itself after rotation.
• Order of rotational symmetry – number of times a figure matches itself during a complete 360° rotation.

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Key Takeaways

• A reflection flips a figure across a line.
• A rotation turns a figure around a fixed point.
• Reflections and rotations preserve size and shape.
• They also preserve side lengths and angle measures.
• A reflection reverses orientation.
• A rotation preserves orientation.
• Reflection across the x-axis:

(x, y) → (x, −y)

• Reflection across the y-axis:

(x, y) → (−x, y)

• Reflection across y = x:

(x, y) → (y, x)

• 90° counterclockwise rotation about the origin:

(x, y) → (−y, x)

• 180° rotation about the origin:

(x, y) → (−x, −y)

• 270° counterclockwise rotation about the origin:

(x, y) → (y, −x)

• Every point and its reflected image are equally distant from the line of reflection.
• Every point maintains its distance from the center during a rotation.
• Reflectional symmetry involves a line.
• Rotational symmetry involves a center and an angle.
• Reflections and rotations are examples of rigid transformations.

A useful way to distinguish the transformations is:

Translation = slide

Reflection = flip

Rotation = turn

What Is a Scale Drawing?

A scale drawing is a drawing that represents a real object at a different size while keeping all measurements in the same proportion.

A scale drawing may be:

• Smaller than the real object.
• Larger than the real object.
• Occasionally the same size.

Scale drawings are useful when the actual object is too large or too small to draw conveniently.

Examples include:

• Maps.
• Building plans.
• Blueprints.
• Model vehicles.
• Floor plans.
• Engineering diagrams.

---

Scale

The scale tells us how measurements on a drawing relate to measurements in real life.

For example:

1 cm represents 5 m

means that every:

1 cm

on the drawing represents:

5 m

in the real object.

If a wall measures 4 cm on the drawing, its actual length is:

4 × 5 = 20 m

---

Why Scale Drawings Are Useful

Imagine trying to draw a school building at its actual size.

That would obviously be impractical.

Instead, we reduce all of its measurements by the same proportion.

Likewise, a very small object can be enlarged so its details are easier to see.

The important idea is that the object's proportions remain unchanged.

---

Scale Factors

A scale factor tells us how much a figure has been enlarged or reduced.

The scale factor can be calculated using:

scale factor = new length ÷ original length

For example, suppose a line that was originally:

4 cm

becomes:

12 cm

The scale factor is:

12 ÷ 4 = 3

Therefore:

scale factor = 3

Every length has been multiplied by 3.

---

Enlargements

An enlargement makes a figure larger while keeping the same proportions.

If the scale factor is:

greater than 1

the figure is enlarged.

For example:

scale factor = 2

means every length is doubled.

A rectangle measuring:

4 cm × 3 cm

becomes:

8 cm × 6 cm

because:

4 × 2 = 8

and:

3 × 2 = 6

---

Reductions

A reduction makes a figure smaller while keeping the same proportions.

If the scale factor is:

between 0 and 1

the figure is reduced.

For example:

scale factor = 0.5

means every length becomes half as long.

A rectangle measuring:

10 cm × 6 cm

becomes:

5 cm × 3 cm

because:

10 × 0.5 = 5

and:

6 × 0.5 = 3

---

Scale Factor of 1

If:

scale factor = 1

the figure remains the same size.

For example:

8 cm × 1 = 8 cm

The figure has not been enlarged or reduced.

---

Using Scale Factors

To calculate a new length:

new length = original length × scale factor

Example

A triangle has a side measuring:

7 cm

It is enlarged using a scale factor of:

3

Therefore:

new length = 7 × 3

new length = 21 cm

---

Finding the Original Length

Sometimes the new measurement and scale factor are known.

Use:

original length = new length ÷ scale factor

Example

A figure has been enlarged by a scale factor of:

4

A side of the enlarged figure measures:

28 cm

Original length:

28 ÷ 4 = 7 cm

---

Similar Figures

Two figures are similar when they have:

• The same shape.
• Equal corresponding angles.
• Corresponding side lengths in the same proportion.

Similar figures do not have to be the same size.

For example, two rectangles measuring:

4 cm × 6 cm

and:

8 cm × 12 cm

are similar.

Each side of the second rectangle is:

2 times

the corresponding side of the first.

---

Similar Is Not the Same as Congruent

Congruent figures have:

• The same shape.
• The same size.

Similar figures have:

• The same shape.
• Not necessarily the same size.

All congruent figures are similar.

However, similar figures are not necessarily congruent.

---

Corresponding Sides

Corresponding sides are sides that occupy matching positions in similar figures.

Suppose triangle ABC is similar to triangle DEF.

The corresponding vertices might be:

A ↔ D

B ↔ E

C ↔ F

Therefore, the corresponding sides are:

AB ↔ DE

BC ↔ EF

AC ↔ DF

It is important to match the correct sides before calculating a scale factor.

---

Corresponding Angles

Corresponding angles in similar figures are equal.

For example, if:

∠A = 60°

then its corresponding angle:

∠D = 60°

Scaling changes lengths, but it does not change the angle measurements.

---

Finding a Scale Factor

Suppose two similar triangles have corresponding sides:

small triangle = 5 cm

large triangle = 15 cm

Calculate:

scale factor = new ÷ original

scale factor = 15 ÷ 5

scale factor = 3

The larger triangle is therefore three times the size of the smaller triangle in each linear dimension.

---

Finding Missing Lengths

Similarity allows us to determine unknown measurements.

Suppose two similar rectangles have:

Small rectangle:

width = 4 cm

length = 6 cm

Large rectangle:

width = 10 cm

length = x

First find the scale factor:

10 ÷ 4 = 2.5

Now apply the same scale factor to the length:

x = 6 × 2.5

x = 15 cm

---

Using Proportions

Missing lengths can also be found using proportions.

Suppose two similar triangles have corresponding sides:

4 cm ↔ 10 cm

6 cm ↔ x

Write:

4/10 = 6/x

Cross multiply:

4x = 60

Therefore:

x = 15 cm

This works because corresponding sides of similar figures have equal ratios.

---

Checking Whether Figures Are Similar

Consider two rectangles:

Rectangle A:

4 cm × 6 cm

Rectangle B:

8 cm × 12 cm

Compare corresponding sides:

8 ÷ 4 = 2

12 ÷ 6 = 2

The scale factor is the same.

Therefore, the rectangles are:

similar

---

Example of Figures That Are Not Similar

Consider:

Rectangle A:

4 cm × 6 cm

Rectangle B:

8 cm × 10 cm

Compare corresponding sides:

8 ÷ 4 = 2

but:

10 ÷ 6 ≈ 1.67

The scale factors are different.

Therefore, the rectangles are:

not similar

Simply being the same type of shape does not always make two figures similar.

---

Squares and Similarity

All squares are similar.

Why?

Every square has:

• Four 90° angles.
• Four equal sides.

A larger square is simply a scaled version of a smaller square.

For example:

A 3 cm × 3 cm square and a 12 cm × 12 cm square have scale factor:

12 ÷ 3 = 4

---

Circles and Similarity

All circles are also similar.

A circle can be enlarged or reduced without changing its basic shape.

If the radius doubles, then:

• Radius doubles.
• Diameter doubles.
• Circumference doubles.

The circle remains similar to the original.

---

Similar Triangles

Two triangles can have different sizes but still be similar.

For similar triangles:

• Corresponding angles are equal.
• Corresponding sides are proportional.

For example:

Triangle A has sides:

3 cm, 4 cm, 5 cm

Triangle B has sides:

6 cm, 8 cm, 10 cm

Each side has been multiplied by:

2

Therefore, the triangles are similar.

---

Scale Drawings and Units

One important challenge with scale drawings is making sure the units match.

Suppose a map scale is:

1 cm : 5 km

A road measures:

7 cm

on the map.

Actual distance:

7 × 5 = 35 km

The answer should be written in:

kilometres

because that is the real-world unit given by the scale.

---

Ratio Scales

Scale can also be written as a ratio.

For example:

1 : 100

This means:

1 unit on the drawing represents 100 of the same units in reality.

Therefore:

1 cm on drawing = 100 cm in reality

Since:

100 cm = 1 m

a 5 cm line on the drawing represents:

500 cm = 5 m

in reality.

---

Understanding 1 : 50

A scale of:

1 : 50

means the real object is 50 times larger than the drawing.

If a wall measures:

8 cm

on the drawing:

actual length = 8 × 50

= 400 cm

Convert:

400 cm = 4 m

Therefore, the actual wall is:

4 m long

---

Understanding 1 : 100

A scale of:

1 : 100

is particularly convenient.

It means:

1 cm on drawing = 100 cm in reality

Since:

100 cm = 1 m

we can also say:

1 cm on drawing = 1 m in reality

Therefore, a 7.5 cm wall on a 1:100 floor plan represents:

7.5 m

---

Finding a Drawing Length

Suppose a building is:

30 m long

and the scale is:

1 cm : 5 m

Drawing length:

30 ÷ 5 = 6 cm

Therefore, the building should be drawn:

6 cm long

---

Finding an Actual Length

Suppose a road measures:

9 cm

on a map with scale:

1 cm : 4 km

Actual length:

9 × 4 = 36 km

---

Finding the Scale

Sometimes the drawing and actual measurements are known.

Suppose:

2 cm on a drawing represents 10 m

Divide both values by 2:

1 cm represents 5 m

Therefore, the scale can be written:

1 cm : 5 m

---

Scale Drawings of Rooms

Architects and designers often use scale drawings of rooms and buildings.

Suppose a classroom measures:

8 m × 6 m

A drawing uses the scale:

1 cm : 1 m

The drawing dimensions will be:

8 cm × 6 cm

Furniture and other objects can then be placed proportionally on the plan.

---

Maps

Maps are scale drawings of real geographic areas.

A map might use:

1 cm : 20 km

Suppose two cities are:

6.5 cm apart

on the map.

Actual distance:

6.5 × 20

= 130 km

Therefore, the map represents a distance of approximately:

130 km

---

Blueprints

A blueprint or technical plan represents an object or structure using accurate scaled measurements.

Blueprints are used for:

• Buildings.
• Machines.
• Bridges.
• Furniture.
• Electrical systems.
• Engineering projects.

Engineers and builders can determine actual measurements from the drawing's scale.

---

Models

Models also use scale.

For example, a model car might have a scale of:

1 : 20

This means the real car is:

20 times larger

in each linear dimension.

If the model car is:

22 cm long

then the real car is:

22 × 20 = 440 cm

or:

4.4 m

long.

---

Scale Model Aircraft

Suppose a model airplane uses a scale of:

1 : 50

The real airplane has a wingspan of:

30 m

Convert to centimetres:

30 m = 3000 cm

Then divide by 50:

3000 ÷ 50 = 60 cm

The model should have a wingspan of:

60 cm

---

Enlarging Small Objects

Scale drawings can also make very small objects easier to examine.

For example, diagrams of:

• Cells.
• Insects.
• Electronic components.
• Machine parts.

may be much larger than the actual objects.

If a drawing uses a scale factor of:

10

then every length shown is 10 times the corresponding actual length.

---

Scale Factor and Perimeter

When all lengths are multiplied by a scale factor, the perimeter changes by the same factor.

Suppose a rectangle has perimeter:

20 cm

and is enlarged by scale factor:

3

New perimeter:

20 × 3 = 60 cm

This happens because every side length has been multiplied by 3.

---

Scale Factor and Area

Area behaves differently.

Suppose a square has sides:

2 cm

Its area is:

2 × 2 = 4 cm²

Enlarge it using scale factor:

3

New side:

2 × 3 = 6 cm

New area:

6 × 6 = 36 cm²

Compare:

36 ÷ 4 = 9

The length scale factor was:

3

but the area scale factor was:

3² = 9

Therefore:

area scale factor = (length scale factor)²

---

Why Area Changes Faster

Imagine a square enlarged by scale factor 2.

Original:

3 cm × 3 cm

Area:

9 cm²

Enlarged:

6 cm × 6 cm

Area:

36 cm²

The side lengths doubled, but:

36 ÷ 9 = 4

The area became four times larger.

This is because:

2² = 4

---

Scale Factor and Volume

For three-dimensional similar objects, volume changes even faster.

If the length scale factor is:

k

then:

volume scale factor = k³

For example, if every dimension doubles:

volume scale factor = 2³ = 8

The new object has eight times the volume.

---

Comparing Linear, Area, and Volume Scale Factors

If the linear scale factor is:

k

then:

For example, if:

k = 3

then:

• Lengths become 3 times larger.
• Perimeter becomes 3 times larger.
• Area becomes 9 times larger.
• Volume becomes 27 times larger.

---

A Reliable Scale Drawing Strategy

When solving a scale drawing problem:

Step 1: Identify the scale.

Step 2: Identify whether you are finding the drawing measurement or actual measurement.

Step 3: Make sure the units are compatible.

Step 4: Multiply or divide using the scale.

Step 5: Include the correct units.

Step 6: Check whether the answer is reasonable.

---

A Reliable Similar-Figures Strategy

When working with similar figures:

Step 1: Identify corresponding sides.

Step 2: Find the scale factor using a pair of known corresponding sides.

Step 3: Apply the same scale factor to the unknown side.

Step 4: Check that all corresponding lengths remain proportional.

---

Worked Example: Map

A map uses:

1 cm : 8 km

Two towns are:

5.5 cm apart

on the map.

Actual distance:

5.5 × 8

= 44 km

Therefore:

The towns are 44 km apart.

---

Worked Example: Blueprint

A blueprint uses a scale of:

1 : 50

A door measures:

4 cm

on the blueprint.

Actual measurement:

4 × 50 = 200 cm

Convert:

200 cm = 2 m

Therefore, the actual door is:

2 m

high or wide, depending on the measurement shown.

---

Worked Example: Similar Triangles

Two triangles are similar.

The smaller triangle has corresponding sides:

4 cm and 7 cm

The larger triangle has:

12 cm and x

Find the scale factor:

12 ÷ 4 = 3

Therefore:

x = 7 × 3

x = 21 cm

---

Worked Example: Reduction

A photograph measures:

24 cm × 18 cm

It is reduced using scale factor:

0.5

New length:

24 × 0.5 = 12 cm

New width:

18 × 0.5 = 9 cm

Therefore, the reduced photograph measures:

12 cm × 9 cm

The proportions have not changed.

---

Common Mistakes

Using Different Scale Factors for Different Sides

For similar figures, the same scale factor must apply to every corresponding length.

If one side doubles while another triples, the figures are not similar.

---

Matching the Wrong Sides

Always identify corresponding sides before calculating.

A long side should correspond to the matching long side, not automatically to whichever measurement appears first.

---

Multiplying When You Should Divide

If you know the drawing measurement and need the larger real measurement, you will often multiply.

If you know the real measurement and need the smaller drawing measurement, you will often divide.

Think about whether your answer should become larger or smaller.

---

Ignoring Units

Consider:

1 cm : 5 m

This does not mean:

1 : 5

because the units are different.

If writing a unitless ratio, convert to the same units first.

Since:

5 m = 500 cm

the ratio is:

1 : 500

---

Assuming Similar Means Congruent

Similar figures have the same:

shape

but may have different:

sizes

Congruent figures have both the same size and shape.

---

Changing Angles During Scaling

Enlargements and reductions change lengths.

They do not change corresponding angle measurements.

---

Applying the Length Scale Factor Directly to Area

If lengths are multiplied by 3, area is not multiplied by 3.

It is multiplied by:

3² = 9

---

Forgetting to Check Reasonableness

If a map uses:

1 cm : 10 km

and a 4 cm distance is calculated as 0.4 km, something has gone wrong.

We expect the real distance to be much larger than the map measurement.

---

Check Your Understanding

1. Define a scale drawing.

2. What information does a scale provide?

3. Define scale factor.

4. What happens when the scale factor is greater than 1?

5. What happens when the scale factor is between 0 and 1?

6. What happens when the scale factor is exactly 1?

7. A 5 cm side is enlarged using scale factor 3. Find the new length.

8. A 12 cm side is reduced using scale factor 0.5. Find the new length.

9. A side changes from 4 cm to 20 cm. Find the scale factor.

10. A side becomes 24 cm after an enlargement with scale factor 3. Find its original length.

11. Define similar figures.

12. What happens to corresponding angles in similar figures?

13. What happens to corresponding side lengths in similar figures?

14. Explain the difference between similar and congruent figures.

15. Are all squares similar? Explain.

16. Are all rectangles similar? Explain.

17. Two similar figures have corresponding sides of 3 cm and 12 cm. Find the scale factor from the smaller figure to the larger figure.

18. A second side measures 5 cm on the smaller figure. Find its corresponding length on the larger figure from Question 17.

19. A map uses the scale 1 cm : 6 km. Two locations are 8 cm apart. Find their represented distance.

20. On the same map, two locations are 42 km apart. How far apart should they be drawn?

21. A blueprint uses a scale of 1:100. A wall measures 7 cm on the plan. Find its actual length in metres.

22. A model car uses a scale of 1:25. The model is 18 cm long. Find the length of the real car.

23. A real building is 40 m long. A plan uses 1 cm : 5 m. How long should the building appear on the plan?

24. Two rectangles measure 4 cm × 6 cm and 10 cm × 15 cm. Are they similar? Explain.

25. Two rectangles measure 4 cm × 6 cm and 8 cm × 10 cm. Are they similar? Explain.

26. A figure is enlarged using scale factor 4. By what factor does its perimeter increase?

27. A figure is enlarged using scale factor 4. By what factor does its area increase?

28. A three-dimensional model is enlarged using scale factor 2. By what factor does its volume increase?

29. Explain why scale drawings are useful for architects and engineers.

30. Give three real-world situations in which scale drawings or similar figures are useful.

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Key Terms

• Scale drawing – representation of an object in which all measurements are changed proportionally.
• Scale – relationship between a measurement on a drawing and the corresponding actual measurement.
• Scale factor – number by which corresponding lengths are multiplied.
• Enlargement – proportional increase in the size of a figure.
• Reduction – proportional decrease in the size of a figure.
• Similar figures – figures with the same shape, equal corresponding angles, and proportional corresponding sides.
• Congruent figures – figures with the same size and shape.
• Corresponding sides – sides that occupy matching positions in similar figures.
• Corresponding angles – angles that occupy matching positions in similar figures.
• Proportion – equality between two ratios.
• Blueprint – technical scale drawing used in design or construction.
• Model – representation of an object, often produced using a particular scale.

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Key Takeaways

• A scale drawing represents a real object at a proportional size.
• The scale connects measurements on the drawing with actual measurements.
• A scale factor describes how much lengths have been multiplied.
• A scale factor greater than 1 produces an enlargement.
• A scale factor between 0 and 1 produces a reduction.
• Similar figures have the same shape but may have different sizes.
• Corresponding angles of similar figures are equal.
• Corresponding sides of similar figures are proportional.
• The same scale factor applies to all corresponding lengths.
• Scale drawings can be used to calculate unknown real-world measurements.
• Units must be considered carefully when using scales.
• Maps, blueprints, floor plans, diagrams, and models all use scale.
• When the length scale factor is k, the perimeter scale factor is also k.
• The area scale factor is k².
• For similar three-dimensional objects, the volume scale factor is k³.

The central relationship to remember is:

new length = original length × scale factor

and for similar figures:

corresponding lengths must all change by the same scale factor.