Geometric Transformations

Translations on the Coordinate Plane

In geometry, a translation means moving a figure from one place to another without turning it, flipping it, or changing its size. You can think of it as a slide.

When a shape is translated, every point on the shape moves the same distance in the same direction. The shape keeps its size and its orientation. That means the translated figure looks exactly like the original figure, just in a different location.

On a coordinate plane, we describe translations by telling how far a point moves left or right and up or down.

Key idea: A translation changes the location of a figure, but not its shape, side lengths, angle measures, or size.

Important vocabulary

• Pre-image: the original figure before the translation
• Image: the new figure after the translation
• Coordinate rule: a rule that tells how each point moves

We often label the image using prime symbols. For example, if the original point is \(A\), the translated point is written as \(A'\).

How translations change coordinates

If a point moves:

• right, add to the \(x\)-coordinate
• left, subtract from the \(x\)-coordinate
• up, add to the \(y\)-coordinate
• down, subtract from the \(y\)-coordinate

The general translation rule is:

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

Here, \(a\) tells how far the point moves left or right, and \(b\) tells how far it moves up or down.

• If \(a\) is positive, move right.
• If \(a\) is negative, move left.
• If \(b\) is positive, move up.
• If \(b\) is negative, move down.

For example:

• \((x, y) \rightarrow (x+3, y)\) means move 3 units right.
• \((x, y) \rightarrow (x-2, y)\) means move 2 units left.
• \((x, y) \rightarrow (x, y+5)\) means move 5 units up.
• \((x, y) \rightarrow (x, y-4)\) means move 4 units down.
• \((x, y) \rightarrow (x+2, y-3)\) means move 2 units right and 3 units down.

How to translate a figure

1. Look at the translation rule or direction.
2. Apply the same move to each vertex of the figure.
3. Plot the new points.
4. Connect the points in the same order as the original figure.

Because every point moves the same way, the image will be congruent to the pre-image. That means the figures are the same size and shape.

Worked Example 1: Translate one point

Point \(A(2, 3)\) is translated 4 units right and 2 units up. Find \(A'\).

Step 1: Moving 4 units right means add 4 to the \(x\)-coordinate.

$$2+4=6$$

Step 2: Moving 2 units up means add 2 to the \(y\)-coordinate.

$$3+2=5$$

So the image is:

$$A'(6, 5)$$

Coordinate rule:

$$ (x, y) \rightarrow (x+4, y+2) $$

Worked Example 2: Translate a triangle

Triangle \(ABC\) has vertices \(A(1,1)\), \(B(4,1)\), and \(C(2,3)\). Translate the triangle by the rule

$$ (x, y) \rightarrow (x-3, y+2) $$

This means move every point 3 units left and 2 units up.

Find each image point:

• \(A(1,1) \rightarrow A'(1-3, 1+2)=(-2,3)\)
• \(B(4,1) \rightarrow B'(4-3, 1+2)=(1,3)\)
• \(C(2,3) \rightarrow C'(2-3, 3+2)=(-1,5)\)

So the translated triangle has vertices:

$$A'(-2,3),\quad B'(1,3),\quad C'(-1,5)$$

Notice that the triangle did not change shape or size. It only slid to a new location.

Worked Example 3: Find the translation rule

Suppose point \(P(-4, 2)\) is translated to \(P'(1, -3)\). Find the translation rule.

Step 1: Compare the \(x\)-coordinates.

$$1-(-4)=5$$

The point moved 5 units right.

Step 2: Compare the \(y\)-coordinates.

$$-3-2=-5$$

The point moved 5 units down.

So the translation rule is:

$$ (x, y) \rightarrow (x+5, y-5) $$

Worked Example 4: Translate a rectangle with negative coordinates

Rectangle \(QRST\) has vertices \(Q(-3,-1)\), \(R(1,-1)\), \(S(1,-4)\), and \(T(-3,-4)\). Translate it 2 units left and 6 units up.

The rule is:

$$ (x, y) \rightarrow (x-2, y+6) $$

Now apply the rule to each vertex:

• \(Q(-3,-1) \rightarrow Q'(-5,5)\)
• \(R(1,-1) \rightarrow R'(-1,5)\)
• \(S(1,-4) \rightarrow S'(-1,2)\)
• \(T(-3,-4) \rightarrow T'(-5,2)\)

So the image is:

$$Q'(-5,5),\quad R'(-1,5),\quad S'(-1,2),\quad T'(-5,2)$$

How to tell if your answer makes sense

• Did every point move the same number of units?
• Did the figure keep the same shape and size?
• Did you add or subtract correctly?
• Did left/right change the \(x\)-coordinate and up/down change the \(y\)-coordinate?

Common mistakes to avoid

• Mixing up \(x\) and \(y\): left and right affect \(x\), not \(y\).
• Using the wrong sign: left and down mean subtract; right and up mean add.
• Moving only one point: every vertex must be translated.
• Changing the shape: a translation does not stretch, shrink, rotate, or reflect the figure.

Quick practice ideas

• Translate \((3,-2)\) by 1 unit left and 4 units up.
• Use the rule \((x,y) \rightarrow (x+2,y-3)\) on point \((-1,5)\).
• Find the rule that maps \((2,7)\) to \((-3,4)\).

Brief Summary

A translation is a slide of a figure on the coordinate plane. Every point moves the same distance in the same direction. Use the rule \((x,y) \rightarrow (x+a,y+b)\) to show how coordinates change: add or subtract from \(x\) for left/right and from \(y\) for up/down. After a translation, the figure keeps the same size and shape.

Reflections on the Coordinate Plane

A reflection is a transformation that flips a figure over a line. The line you flip over is called the line of reflection.

When a figure is reflected, it keeps the same size and shape. This means a reflection is a rigid transformation. Only the position and orientation change.

In 8th grade, you will often reflect points and shapes across these lines:

• the x-axis
• the y-axis
• the line \(y=x\)
• the line \(y=-x\)

The key idea is that each point and its image are the same distance from the line of reflection, but on opposite sides.

Important vocabulary:

• Pre-image: the original figure
• Image: the figure after the reflection
• Prime notation: the image is often labeled with a prime, such as \(A'\)

1. Reflecting across the x-axis

When you reflect a point across the x-axis, the x-coordinate stays the same, but the y-coordinate changes sign.

The rule is:

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

This makes sense because a point above the x-axis moves the same distance below it, or a point below moves the same distance above it.

Example:

If \(A=(4,3)\), then reflecting across the x-axis gives:

$$ A'=(4,-3) $$

If \(B=(-2,-5)\), then reflecting across the x-axis gives:

$$ B'=(-2,5) $$

2. Reflecting across the y-axis

When you reflect a point across the y-axis, the y-coordinate stays the same, but the x-coordinate changes sign.

The rule is:

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

Example:

If \(C=(6,-1)\), then reflecting across the y-axis gives:

$$ C'=(-6,-1) $$

If \(D=(-3,4)\), then reflecting across the y-axis gives:

$$ D'=(3,4) $$

3. Reflecting across the line \(y=x\)

The line \(y=x\) is a diagonal line that goes through the origin and rises from left to right.

When you reflect a point across \(y=x\), the coordinates switch places.

The rule is:

$$ (x,y) \rightarrow (y,x) $$

Example:

If \(E=(2,7)\), then reflecting across \(y=x\) gives:

$$ E'=(7,2) $$

If \(F=(-4,1)\), then reflecting across \(y=x\) gives:

$$ F'=(1,-4) $$

4. Reflecting across the line \(y=-x\)

The line \(y=-x\) is another diagonal line through the origin, but it falls from left to right.

When you reflect a point across \(y=-x\), the coordinates switch places and both signs change.

The rule is:

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

Example:

If \(G=(3,5)\), then reflecting across \(y=-x\) gives:

$$ G'=(-5,-3) $$

If \(H=(-2,6)\), then reflecting across \(y=-x\) gives:

$$ H'=(-6,2) $$

A quick chart of reflection rules

• Across the x-axis: \((x,y) \rightarrow (x,-y)\)
• Across the y-axis: \((x,y) \rightarrow (-x,y)\)
• Across \(y=x\): \((x,y) \rightarrow (y,x)\)
• Across \(y=-x\): \((x,y) \rightarrow (-y,-x)\)

How to tell if your reflection makes sense

• If you reflect across the x-axis, only the point’s vertical position should change.
• If you reflect across the y-axis, only the point’s horizontal position should change.
• If you reflect across \(y=x\), the coordinates should trade places.
• If you reflect across \(y=-x\), the coordinates should trade places and change signs.

Worked Example 1: Reflecting one point across the x-axis

Reflect the point \(P=(-5,2)\) across the x-axis.

Step 1: Use the rule for the x-axis:

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

Step 2: Keep the x-coordinate the same and change the sign of the y-coordinate.

$$ (-5,2) \rightarrow (-5,-2) $$

Answer: \(P'=(-5,-2)\)

Worked Example 2: Reflecting one point across \(y=x\)

Reflect the point \(Q=(8,-3)\) across the line \(y=x\).

Step 1: Use the rule:

$$ (x,y) \rightarrow (y,x) $$

Step 2: Switch the coordinates.

$$ (8,-3) \rightarrow (-3,8) $$

Answer: \(Q'=(-3,8)\)

Worked Example 3: Reflecting a triangle across the y-axis

Triangle \(ABC\) has vertices:

• \(A=(1,2)\)
• \(B=(4,5)\)
• \(C=(3,1)\)

Reflect triangle \(ABC\) across the y-axis.

Step 1: Use the rule for reflection across the y-axis:

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

Step 2: Reflect each vertex.

• \(A=(1,2) \rightarrow A'=(-1,2)\)
• \(B=(4,5) \rightarrow B'=(-4,5)\)
• \(C=(3,1) \rightarrow C'=(-3,1)\)

Answer: The reflected triangle has vertices

$$ A'=(-1,2), \quad B'=(-4,5), \quad C'=(-3,1) $$

Worked Example 4: Reflecting a shape across \(y=-x\)

Rectangle \(JKLM\) has vertices:

• \(J=(2,1)\)
• \(K=(5,1)\)
• \(L=(5,-2)\)
• \(M=(2,-2)\)

Reflect the rectangle across the line \(y=-x\).

Step 1: Use the rule:

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

Step 2: Reflect each point.

• \(J=(2,1) \rightarrow J'=(-1,-2)\)
• \(K=(5,1) \rightarrow K'=(-1,-5)\)
• \(L=(5,-2) \rightarrow L'=(2,-5)\)
• \(M=(2,-2) \rightarrow M'=(2,-2)\)

Answer:

$$ J'=(-1,-2), \quad K'=(-1,-5), \quad L'=(2,-5), \quad M'=(2,-2) $$

Notice that \(M=(2,-2)\) stayed the same after the reflection. That is because the point lies on the line \(y=-x\). Any point on the line of reflection does not move.

Common mistakes to avoid

• Mixing up the x-axis and y-axis rules. Remember: x-axis changes the y; y-axis changes the x.
• Forgetting to switch coordinates when reflecting across \(y=x\) or \(y=-x\).
• Forgetting the sign changes for reflection across \(y=-x\).
• Only reflecting one point of a figure. Every vertex must be reflected.

Helpful memory tips

• x-axis: think “flip up and down,” so the y changes.
• y-axis: think “flip left and right,” so the x changes.
• \(y=x\): just swap the coordinates.
• \(y=-x\): swap the coordinates and make them opposites.

Summary

A reflection flips a figure across a line while keeping the same size and shape. On the coordinate plane, you can use rules to reflect points quickly and accurately.

Use these four main rules:

• $$ (x,y) \rightarrow (x,-y) $$ across the x-axis
• $$ (x,y) \rightarrow (-x,y) $$ across the y-axis
• $$ (x,y) \rightarrow (y,x) $$ across \(y=x\)
• $$ (x,y) \rightarrow (-y,-x) $$ across \(y=-x\)

If you remember what changes, what stays the same, and when coordinates switch, you will be able to reflect both points and whole figures with confidence.

Rotations About the Origin are a type of geometric transformation on the coordinate plane. A rotation turns a point or a figure around a fixed center. In this lesson, the center of rotation will always be the origin, which is the point \\( (0,0) \\).

When a figure rotates around the origin, its shape and size stay the same. Only its position changes. This means a rotation is a rigid transformation.

In 8th grade, you will usually rotate points and figures by 90°, 180°, or 270°, either clockwise or counterclockwise.

Important idea: A rotation changes the coordinates of each point using a rule. If you know the rule, you can quickly find the image of any point.

Before we begin:

• The original point is called the preimage.
• The new point after rotation is called the image.
• If the original point is \\(A(x,y)\\), the rotated point might be named \\(A'\\) or \\(A''\\), depending on the problem.

Clockwise means turning in the direction of a clock’s hands.

Counterclockwise means turning the opposite way.

Let’s learn the coordinate rules for rotations about the origin.

1. Rotation of 90° counterclockwise

The rule is:

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

This means:

• The old \\(y\\)-value becomes the new \\(x\\)-value, but with the opposite sign.
• The old \\(x\\)-value becomes the new \\(y\\)-value.

2. Rotation of 180°

The rule is:

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

This works for both clockwise and counterclockwise, because turning 180° in either direction lands in the same place.

3. Rotation of 270° counterclockwise

The rule is:

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

4. Rotation of 90° clockwise

The rule is:

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

Notice that this is the same rule as 270° counterclockwise.

5. Rotation of 270° clockwise

The rule is:

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

Notice that this is the same rule as 90° counterclockwise.

Here is a helpful list to remember:

• 90° counterclockwise: \\( (x,y) \rightarrow (-y,x) \\)
• 180°: \\( (x,y) \rightarrow (-x,-y) \\)
• 270° counterclockwise: \\( (x,y) \rightarrow (y,-x) \\)
• 90° clockwise: \\( (x,y) \rightarrow (y,-x) \\)
• 270° clockwise: \\( (x,y) \rightarrow (-y,x) \\)

A useful pattern:

• For 90° and 270° rotations, the coordinates switch places.
• Then one of the signs changes.
• For 180°, the coordinates do not switch places, but both signs change.

Now let’s work through some examples.

Worked Example 1: Rotate one point 90° counterclockwise

Rotate point \\(A(4,1)\\) 90° counterclockwise about the origin.

Use the rule:

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

Substitute \\(x=4\\) and \\(y=1\\):

$$ (4,1) \rightarrow (-1,4) $$

So the image is \\(A'(-1,4)\\).

Check the pattern: the coordinates switched from \\(4,1\\) to \\(1,4\\), and then the new \\(x\\)-value became negative.

Worked Example 2: Rotate one point 180°

Rotate point \\(B(-3,5)\\) 180° about the origin.

Use the rule:

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

Substitute the coordinates:

$$ (-3,5) \rightarrow (3,-5) $$

So the image is \\(B'(3,-5)\\).

What happened? Both signs changed. The point moved to the opposite side of the origin.

Worked Example 3: Rotate one point 90° clockwise

Rotate point \\(C(-2,-6)\\) 90° clockwise about the origin.

Use the rule:

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

Substitute the coordinates:

$$ (-2,-6) \rightarrow (-6,2) $$

So the image is \\(C'(-6,2)\\).

Notice: the coordinates switched places, and the new \\(y\\)-value became the opposite of the original \\(x\\)-value.

Worked Example 4: Rotate a triangle 270° counterclockwise

Triangle \\(ABC\\) has vertices:

• \\(A(1,2)\\)
• \\(B(4,2)\\)
• \\(C(2,5)\\)

Rotate the triangle 270° counterclockwise about the origin.

Use the rule:

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

Now rotate each vertex one at a time.

For \\(A(1,2)\\):

$$ (1,2) \rightarrow (2,-1) $$

So \\(A'(2,-1)\\).

For \\(B(4,2)\\):

$$ (4,2) \rightarrow (2,-4) $$

So \\(B'(2,-4)\\).

For \\(C(2,5)\\):

$$ (2,5) \rightarrow (5,-2) $$

So \\(C'(5,-2)\\).

The rotated triangle has vertices:

• \\(A'(2,-1)\\)
• \\(B'(2,-4)\\)
• \\(C'(5,-2)\\)

How to rotate a whole figure:

1. Look at the direction and angle of rotation.
2. Choose the correct rotation rule.
3. Apply the rule to each vertex.
4. Plot the new points if needed.
5. Connect the points in the same order to make the rotated figure.

How to avoid common mistakes:

• Do not confuse clockwise and counterclockwise. Read the direction carefully.
• Do not forget sign changes. A wrong sign gives the wrong location.
• Do not mix up 90° and 270°. They use different directions unless matched correctly.
• Remember that 180° always gives the same result whether clockwise or counterclockwise.
• Rotate every vertex of a figure. One point is not enough for a whole polygon.

Quick memory tips:

• 90° rotations: switch the coordinates.
• 180° rotation: change both signs.
• 90° counterclockwise: \\((-y,x)\\)
• 90° clockwise: \\((y,-x)\\)

Mini practice ideas:

• Rotate \\((3,-2)\\) by 180°.
• Rotate \\((-5,1)\\) by 90° counterclockwise.
• Rotate \\((2,4)\\) by 90° clockwise.

The answers would be:

• \\((3,-2) \rightarrow (-3,2)\\)
• \\((-5,1) \rightarrow (-1,-5)\\)
• \\((2,4) \rightarrow (4,-2)\\)

Summary

A rotation about the origin turns a point or figure around \\( (0,0) \\) without changing its size or shape. To rotate points, use the correct coordinate rule based on the angle and direction. The most important rules to remember are:

• \\( (x,y) \rightarrow (-y,x) \\) for 90° counterclockwise
• \\( (x,y) \rightarrow (-x,-y) \\) for 180°
• \\( (x,y) \rightarrow (y,-x) \\) for 90° clockwise

If you practice these rules and pay close attention to signs, you will be able to rotate both points and entire figures correctly.

Properties of Rigid Motions

In geometry, a transformation is a change in the position of a figure. Some transformations slide, flip, or turn a figure without changing its size or shape.

These special transformations are called rigid motions. A rigid motion keeps the figure congruent to the original figure. That means the image has the same size and the same shape as the original.

The three rigid motions you will study are:

• Translation: a slide
• Reflection: a flip
• Rotation: a turn

When a figure is transformed, the original figure is called the preimage, and the new figure is called the image.

The most important idea in this lesson is that rigid motions preserve certain properties. In math, preserve means keep unchanged.

Rigid motions preserve:

• segment length
• angle measure
• parallel lines

This means if a segment is 5 units long before a rigid motion, it is still 5 units long after. If an angle measures \(60^\circ\) before, it still measures \(60^\circ\) after. If two lines are parallel before, they are still parallel after.

Why does this matter? Because it helps us prove that two figures are congruent and helps us understand how shapes move on the coordinate plane.

1. Translation

A translation slides every point of a figure the same distance in the same direction.

For example, moving a point right 3 units and up 2 units follows the rule

$$ (x,y) \rightarrow (x+3, y+2) $$

In a translation:

• lengths stay the same
• angles stay the same
• parallel lines stay parallel
• the orientation stays the same

If triangle \(ABC\) is translated to triangle \(A'B'C'\), then:

• \(AB = A'B'\)
• \(BC = B'C'\)
• \(\angle A = \angle A'\)
• Any parallel sides remain parallel

2. Reflection

A reflection flips a figure across a line called the line of reflection.

Some common reflection rules are:

• Across the x-axis: $$ (x,y) \rightarrow (x,-y) $$
• Across the y-axis: $$ (x,y) \rightarrow (-x,y) $$

In a reflection:

• lengths stay the same
• angles stay the same
• parallel lines stay parallel
• the figure may face the opposite direction

Even though the figure looks flipped, the side lengths and angle measures do not change.

3. Rotation

A rotation turns a figure around a fixed point called the center of rotation.

On a coordinate plane, many problems use rotations around the origin.

Common rotation rules are:

• \(90^\circ\) counterclockwise: $$ (x,y) \rightarrow (-y,x) $$
• \(180^\circ\): $$ (x,y) \rightarrow (-x,-y) $$
• \(270^\circ\) counterclockwise: $$ (x,y) \rightarrow (y,-x) $$

In a rotation:

• lengths stay the same
• angles stay the same
• parallel lines stay parallel
• the figure turns, but does not change size or shape

What does “preserve segment length” mean?

If a segment has endpoints \(A\) and \(B\), then after a rigid motion the image segment \(A'B'\) has the same length.

So if \(AB = 7\), then \(A'B' = 7\) too.

This is true for translations, reflections, and rotations.

What does “preserve angle measure” mean?

If an angle measures \(35^\circ\), then after a rigid motion its image also measures \(35^\circ\).

So if \(m\angle P = 35^\circ\), then \(m\angle P' = 35^\circ\).

What does “preserve parallel lines” mean?

If two lines never meet before a rigid motion, they still never meet after the motion.

For example, if \(\overline{AB}\) is parallel to \(\overline{CD}\), written \(\overline{AB} \parallel \overline{CD}\), then after a rigid motion:

$$ \overline{A'B'} \parallel \overline{C'D'} $$

How can students verify this experimentally?

To verify experimentally means to check by drawing, measuring, or using coordinates.

You can:

• plot a figure on a coordinate plane
• perform a translation, reflection, or rotation
• measure side lengths before and after
• measure angles before and after
• check whether parallel sides are still parallel

If the measurements match, you have verified that the rigid motion preserved those properties.

Worked Example 1: Translation preserves length

Segment \(AB\) has endpoints \(A(1,2)\) and \(B(4,2)\). Translate it using the rule

$$ (x,y) \rightarrow (x+2,y+3) $$

Step 1: Find the image points.

• \(A(1,2) \rightarrow A'(3,5)\)
• \(B(4,2) \rightarrow B'(6,5)\)

Step 2: Compare lengths.

Segment \(AB\) is horizontal. Its length is the difference in the x-values:

$$ 4 - 1 = 3 $$

Segment \(A'B'\) is also horizontal. Its length is:

$$ 6 - 3 = 3 $$

Conclusion: The translation preserved segment length because \(AB = A'B' = 3\).

Worked Example 2: Reflection preserves angle measure

Suppose triangle \(ABC\) has an angle \(\angle A = 50^\circ\). The triangle is reflected across the y-axis to form triangle \(A'B'C'\).

What is \(m\angle A'\)?

A reflection is a rigid motion, so it preserves angle measure.

Therefore,

$$ m\angle A' = 50^\circ $$

Conclusion: The reflected angle has the same measure as the original angle.

Worked Example 3: Rotation preserves parallel lines

A quadrilateral has opposite sides \(\overline{PQ}\) and \(\overline{RS}\) that are parallel.

The figure is rotated \(90^\circ\) around the origin to form \(P'Q'R'S'\).

Are \(\overline{P'Q'}\) and \(\overline{R'S'}\) parallel?

Yes. A rotation is a rigid motion, and rigid motions preserve parallel lines.

So if

$$ \overline{PQ} \parallel \overline{RS} $$

then

$$ \overline{P'Q'} \parallel \overline{R'S'} $$

Conclusion: The rotated image still has those sides parallel.

Worked Example 4: Using coordinates to verify a rotation

Triangle \(ABC\) has points \(A(1,1)\), \(B(3,1)\), and \(C(2,4)\). Rotate the triangle \(180^\circ\) about the origin.

The rule for a \(180^\circ\) rotation is

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

Step 1: Find the image points.

• \(A(1,1) \rightarrow A'(-1,-1)\)
• \(B(3,1) \rightarrow B'(-3,-1)\)
• \(C(2,4) \rightarrow C'(-2,-4)\)

Step 2: Check one side length.

Segment \(AB\) is horizontal from \(x=1\) to \(x=3\), so its length is 2.

Segment \(A'B'\) is horizontal from \(x=-1\) to \(x=-3\), so its length is also 2.

Step 3: Check an angle idea.

Since a rotation is a rigid motion, all angles in the image triangle match the angles in the original triangle.

Conclusion: The rotation preserved length and angle measure, so the image triangle is congruent to the original triangle.

Important idea: Rigid motions do not change size

A rigid motion keeps distances the same. That is why the figure stays congruent.

This is different from a dilation, which can change the size of a figure. A dilation is not a rigid motion unless the scale factor is 1.

How to recognize that a transformation is a rigid motion

• If the figure is only slid, flipped, or turned, it is a rigid motion.
• If the side lengths stay the same, it may be a rigid motion.
• If the angle measures stay the same, that supports that it is a rigid motion.
• If the figure becomes larger or smaller, it is not a rigid motion.

Quick comparison

• Translation: preserves length, angle measure, and parallel lines
• Reflection: preserves length, angle measure, and parallel lines
• Rotation: preserves length, angle measure, and parallel lines

Common mistakes to avoid

• Thinking a reflected figure has different angle measures because it looks backwards
• Forgetting that a rotation changes position, not size
• Mixing up rigid motions with dilations
• Checking only one point instead of the whole figure

Summary

Rigid motions are translations, reflections, and rotations. They move a figure without changing its size or shape.

These transformations preserve segment length, angle measure, and parallel lines. That means the image is congruent to the original figure.

You can verify these properties experimentally by plotting figures, applying a transformation, and checking measurements before and after. If the lengths, angles, and parallel sides stay the same, then the transformation is a rigid motion.

Dilations and Scale Factor

In geometry, a dilation is a transformation that changes the size of a figure but keeps its shape the same. That means the new figure is similar to the original figure.

When a figure is dilated, every point moves away from or toward a fixed point called the center of dilation. The amount of change is controlled by a number called the scale factor.

This lesson will help you understand what dilations do, how to use the scale factor, and how to find the coordinates of a dilated figure on the coordinate plane.

1. What is a dilation?

A dilation makes a figure either larger or smaller.

• If the figure gets larger, it is an enlargement.
• If the figure gets smaller, it is a reduction.

A dilation does not change angle measures. It also keeps corresponding sides in the same proportion. Because of this, the original figure and the image are similar figures.

2. What is the scale factor?

The scale factor tells how much each length changes in a dilation.

• If the scale factor is greater than 1, the image is larger.
• If the scale factor is between 0 and 1, the image is smaller.
• If the scale factor is 1, the figure stays the same size.

For example:

• Scale factor of 2 means every length is doubled.
• Scale factor of 3 means every length is tripled.
• Scale factor of \(\frac{1}{2}\) means every length becomes half as long.
• Scale factor of \(\frac{1}{4}\) means every length becomes one fourth as long.

If a side length is 6 units and the scale factor is 2, the new length is

$$6 \times 2 = 12$$

If a side length is 6 units and the scale factor is \(\frac{1}{2}\), the new length is

$$6 \times \frac{1}{2} = 3$$

3. Center of dilation

The center of dilation is the point from which the figure grows or shrinks. Every point on the figure moves along a straight path starting from the center of dilation.

In many coordinate plane problems in 8th Grade, the center of dilation is the origin, which is \((0,0)\).

When the center of dilation is the origin, the rule is simple:

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

Here, \(k\) is the scale factor.

This means:

• Multiply the x-coordinate by the scale factor.
• Multiply the y-coordinate by the scale factor.

4. How coordinates change in a dilation

If the center is the origin and the scale factor is 2, then:

$$ (x,y) \rightarrow (2x,2y) $$

If the center is the origin and the scale factor is \(\frac{1}{2}\), then:

$$ (x,y) \rightarrow \left(\frac{x}{2},\frac{y}{2}\right) $$

This rule works for every point in the figure.

5. Worked Example 1: Dilating one point

Dilate point \(A(3,4)\) by a scale factor of 2 from the origin.

Step 1: Use the rule \((x,y) \rightarrow (2x,2y)\).

Step 2: Multiply each coordinate by 2.

$$A(3,4) \rightarrow A'(6,8)$$

Answer: The image of point \(A\) is \(A'(6,8)\).

This makes sense because the point moves farther from the origin, so the image is part of an enlargement.

6. Worked Example 2: Reducing one point

Dilate point \(B(-8,6)\) by a scale factor of \(\frac{1}{2}\) from the origin.

Step 1: Use the rule

$$ (x,y) \rightarrow \left(\frac{x}{2},\frac{y}{2}\right) $$

Step 2: Divide each coordinate by 2.

$$B(-8,6) \rightarrow B'(-4,3)$$

Answer: The image of point \(B\) is \(B'(-4,3)\).

The image is closer to the origin, so this is a reduction.

7. Worked Example 3: Dilating a triangle

Triangle \(ABC\) has vertices \(A(1,2)\), \(B(4,2)\), and \(C(1,5)\). Dilate the triangle by a scale factor of 3 from the origin.

Step 1: Multiply each coordinate by 3.

• \(A(1,2) \rightarrow A'(3,6)\)
• \(B(4,2) \rightarrow B'(12,6)\)
• \(C(1,5) \rightarrow C'(3,15)\)

Answer: The dilated triangle has vertices

$$A'(3,6), \quad B'(12,6), \quad C'(3,15)$$

Notice that each side length becomes 3 times as long. The triangle keeps the same shape, but it becomes larger.

8. Worked Example 4: Finding the scale factor

A segment has length 5 units. After dilation, its image has length 15 units. What is the scale factor?

Use the relationship

$$\text{image length} = \text{original length} \times \text{scale factor}$$

Substitute the values:

$$15 = 5 \times k$$

Solve for \(k\):

$$k = \frac{15}{5} = 3$$

Answer: The scale factor is \(3\).

Because the scale factor is greater than 1, the dilation is an enlargement.

9. How to recognize enlargements and reductions

• If \(k > 1\), the image is bigger.
• If \(0 < k < 1\), the image is smaller.
• If \(k = 1\), there is no size change.

Examples:

• \(k=4\): enlargement
• \(k=\frac{3}{2}\): enlargement
• \(k=\frac{1}{3}\): reduction
• \(k=\frac{3}{4}\): reduction

10. Important ideas to remember

• A dilation changes size, not shape.
• The center of dilation is the point from which all points move.
• The scale factor tells how much the figure grows or shrinks.
• When the center is the origin, multiply both coordinates by the scale factor.
• Corresponding side lengths are multiplied by the same scale factor.

11. Common mistakes to avoid

• Forgetting to multiply both coordinates: In a dilation from the origin, both \(x\) and \(y\) must be multiplied by the scale factor.
• Mixing up enlargement and reduction: A scale factor bigger than 1 enlarges; a scale factor between 0 and 1 reduces.
• Adding instead of multiplying: Dilations use multiplication, not addition.
• Changing the shape: A true dilation keeps the figure similar to the original.

12. Quick practice ideas

Try these on your own:

1. Dilate \((2,5)\) by a scale factor of 4 from the origin.
2. Dilate \((-6,9)\) by a scale factor of \(\frac{1}{3}\) from the origin.
3. A side length of 8 becomes 20 after dilation. Find the scale factor.

Answers:

• \((8,20)\)
• \((-2,3)\)
• \(\frac{20}{8}=\frac{5}{2}\)

Summary

A dilation is a transformation that makes a figure larger or smaller while keeping its shape the same. The amount of change is called the scale factor.

When the center of dilation is the origin, use the rule $$ (x,y) \rightarrow (kx,ky) $$. If \(k>1\), the figure enlarges. If \(0<k<1\), the figure reduces. Understanding this rule will help you graph and describe dilations on the coordinate plane.

Coordinate Rules for Dilations

A dilation is a transformation that changes the size of a figure but keeps its shape the same. When a figure is dilated, the new figure is called the image.

In this lesson, we will learn how to write and use coordinate rules for dilations centered at the origin. The origin is the point \,\((0,0)\).

When a dilation is centered at the origin, every point \,\((x,y)\) is multiplied by the same number, called the scale factor.

The coordinate rule is:

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

Here, \,\(k\) is the scale factor.

• If \,\(k > 1\), the image is an enlargement. The figure gets bigger.
• If \,\(0 < k < 1\), the image is a reduction. The figure gets smaller.
• If \,\(k = 1\), the figure stays the same size.

The important idea is simple: multiply both coordinates by the scale factor.

For example, if the scale factor is \,\(2\), then:

$$ (x,y) \rightarrow (2x,2y) $$

If the scale factor is \,\(\frac{1}{2}\), then:

$$ (x,y) \rightarrow \left(\frac{x}{2},\frac{y}{2}\right) $$

This works because each point moves along the line from the origin to that point. A larger scale factor moves the point farther from the origin. A smaller scale factor moves the point closer to the origin.

How to use the rule

1. Look at the original point \,\((x,y)\).
2. Find the scale factor \,\(k\).
3. Multiply the x-coordinate by \,\(k\).
4. Multiply the y-coordinate by \,\(k\).
5. Write the new point.

Worked Example 1: Dilation with a whole-number scale factor

Dilate the point \,\((3,4)\) by a scale factor of \,\(2\) centered at the origin.

Use the rule:

$$ (x,y) \rightarrow (2x,2y) $$

Substitute \,\(x=3\) and \,\(y=4\):

$$ (3,4) \rightarrow (2\cdot 3, 2\cdot 4) = (6,8) $$

So the image of \,\((3,4)\) is \((6,8)\).

Because the scale factor is greater than 1, the point moves farther from the origin.

Worked Example 2: Dilation with a fraction scale factor

Dilate the point \,\((-8,6)\) by a scale factor of \,\(\frac{1}{2}\) centered at the origin.

Use the rule:

$$ (x,y) \rightarrow \left(\frac{1}{2}x,\frac{1}{2}y\right) $$

Substitute the coordinates:

$$ (-8,6) \rightarrow \left(\frac{1}{2}(-8),\frac{1}{2}(6)\right) $$

$$ (-8,6) \rightarrow (-4,3) $$

So the image is \((-4,3)\).

Because the scale factor is between 0 and 1, the point moves closer to the origin.

Worked Example 3: Dilating a whole figure

Triangle \,\(ABC\) has vertices \,\(A(1,2)\), \,\(B(4,2)\), and \,\(C(2,5)\). Dilate the triangle by a scale factor of \,\(3\) centered at the origin.

Use the rule:

$$ (x,y) \rightarrow (3x,3y) $$

Find each image point one at a time.

For point \,\(A(1,2)\):

$$ A(1,2) \rightarrow A'(3,6) $$

For point \,\(B(4,2)\):

$$ B(4,2) \rightarrow B'(12,6) $$

For point \,\(C(2,5)\):

$$ C(2,5) \rightarrow C'(6,15) $$

So the dilated image is triangle \,\(A'B'C'\) with vertices:

• \(A'(3,6)\)
• \(B'(12,6)\)
• \(C'(6,15)\)

Notice that every coordinate was multiplied by 3.

Worked Example 4: Writing the rule from a scale factor

A dilation centered at the origin has scale factor \,\(\frac{3}{4}\). What is the coordinate rule? Then use it on the point \,\((12,-8)\).

First, write the rule:

$$ (x,y) \rightarrow \left(\frac{3}{4}x,\frac{3}{4}y\right) $$

Now apply it to \,\((12,-8)\):

$$ (12,-8) \rightarrow \left(\frac{3}{4}\cdot 12, \frac{3}{4}\cdot (-8)\right) $$

$$ (12,-8) \rightarrow (9,-6) $$

So the image is \((9,-6)\).

What stays the same in a dilation?

• The figure keeps the same shape.
• Corresponding angles stay the same.
• The image and original lie on the same rays from the origin.

What changes in a dilation?

• The size usually changes.
• The coordinates change based on the scale factor.
• The distance from the origin is multiplied by the scale factor.

Common mistakes to avoid

• Do not add the scale factor. In a dilation, you multiply.
• Make sure you multiply both coordinates.
• If the scale factor is a fraction, the point should usually move closer to the origin.
• Keep track of negative signs. For example, \,\(2\cdot (-3) = -6\).

Quick check

If a point is \,\((5,-2)\) and the scale factor is \,\(4\), then:

$$ (5,-2) \rightarrow (20,-8) $$

If a point is \,\((5,-2)\) and the scale factor is \,\(\frac{1}{5}\), then:

$$ (5,-2) \rightarrow (1,-\frac{2}{5}) $$

Summary

A dilation centered at the origin follows the rule:

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

The number \,\(k\) is the scale factor. If \,\(k\) is greater than 1, the figure gets larger. If \,\(k\) is between 0 and 1, the figure gets smaller. To find the image of a point, multiply both coordinates by \,\(k\).

Successive Transformations means doing more than one transformation in order to a figure on the coordinate plane.

A transformation changes a figure’s position, direction, or size. In 8th Grade, the main transformations are:

• Translation: slide
• Reflection: flip
• Rotation: turn
• Dilation: resize

When transformations happen one after another, the order matters. A figure translated and then reflected may end up in a different place than if it were reflected first and then translated.

In this lesson, you will learn how to follow a sequence of transformations, track coordinates carefully, and sometimes recognize when two or more transformations can be described as one single transformation.

1. Review of basic coordinate rules

It helps to know the common coordinate rules for transformations on the coordinate plane.

• Translation by  vector                                                                     by                     (a,b):     (x,y) \rightarrow (x+a, y+b)
• Reflection across the x-axis: \((x,y) \rightarrow (x,-y)\)
• Reflection across the y-axis: \((x,y) \rightarrow (-x,y)\)
• Reflection across the line \(y=x\): \((x,y) \rightarrow (y,x)\)
• Rotation of \(90^\circ\) counterclockwise about the origin: \((x,y) \rightarrow (-y,x)\)
• Rotation of \(180^\circ\) about the origin: \((x,y) \rightarrow (-x,-y)\)
• Rotation of \(90^\circ\) clockwise about the origin: \((x,y) \rightarrow (y,-x)\)
• Dilation with scale factor \(k\) about the origin: \((x,y) \rightarrow (kx,ky)\)

2. What makes transformations successive?

If a triangle is translated and then reflected, those are successive transformations. The image after the first transformation becomes the starting figure for the second transformation.

You can think of the steps like this:

1. Start with the original figure.
2. Apply the first transformation.
3. Use the new coordinates.
4. Apply the second transformation.
5. Continue if there are more steps.

Important: Do not try to jump to the end too quickly. It is usually easiest to rewrite the coordinates after each transformation.

3. Why order matters

Suppose you start with the point \((1,2)\).

If you reflect across the x-axis first, you get:

\((1,2) \rightarrow (1,-2)\)

Then if you translate right 3 units, you get:

\((1,-2) \rightarrow (4,-2)\)

But if you translate first, then reflect, the result is different.

Translate right 3 units:

\((1,2) \rightarrow (4,2)\)

Then reflect across the x-axis:

\((4,2) \rightarrow (4,-2)\)

In this case, the final answer happened to be the same. But that does not always happen. Let’s change the translation.

Start again with \((1,2)\), and use a translation up 3 units.

Reflect first:

\((1,2) \rightarrow (1,-2)\)

Then translate up 3:

\((1,-2) \rightarrow (1,1)\)

Now reverse the order.

Translate up 3 first:

\((1,2) \rightarrow (1,5)\)

Then reflect across the x-axis:

\((1,5) \rightarrow (1,-5)\)

The final points are \((1,1)\) and \((1,-5)\), which are different. This shows that order matters.

4. A good strategy for successive transformations

When working with a figure, follow these steps:

1. List the original coordinates.
2. Do the first transformation only.
3. Write the new coordinates clearly.
4. Do the next transformation using the new coordinates.
5. Check your work. Make sure each point followed the correct rule.

If the figure has several vertices, organize your work in a table.

For example:

Original point \(A(x,y)\) becomes \(A'(\ ,\ )\) after the first transformation, then \(A''(\ ,\ )\) after the second transformation.

The marks like prime \(( ')\) and double prime \(( '' )\) help show the order:

• Original figure: \(A, B, C\)
• After first transformation: \(A', B', C'\)
• After second transformation: \(A'', B'', C''\)

5. Worked Example 1: translation, then reflection

Triangle \(ABC\) has vertices:

$$A(1,1), \quad B(3,1), \quad C(2,4)$$

First, translate the triangle right 2 units and up 1 unit. Then reflect it across the x-axis.

Step 1: Translation

The rule is:

$$ (x,y) \rightarrow (x+2, y+1) $$

• \(A(1,1) \rightarrow A'(3,2)\)
• \(B(3,1) \rightarrow B'(5,2)\)
• \(C(2,4) \rightarrow C'(4,5)\)

Step 2: Reflect across the x-axis

The rule is:

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

• \(A'(3,2) \rightarrow A''(3,-2)\)
• \(B'(5,2) \rightarrow B''(5,-2)\)
• \(C'(4,5) \rightarrow C''(4,-5)\)

Final image:

$$A''(3,-2), \quad B''(5,-2), \quad C''(4,-5)$$

What should you notice? We did not reflect the original points. We reflected the points after the translation.

6. Worked Example 2: two reflections that make one rotation

Start with point \(P(2, -1)\).

Reflect it across the x-axis, then reflect it across the y-axis.

Step 1: Reflect across the x-axis

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

So:

$$ (2,-1) \rightarrow (2,1) $$

Step 2: Reflect across the y-axis

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

So:

$$ (2,1) \rightarrow (-2,1) $$

Final answer: \((-2,1)\)

Now compare this to a \(180^\circ\) rotation about the origin:

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

If we rotate \((2,-1)\) by \(180^\circ\), we get:

$$ (2,-1) \rightarrow (-2,1) $$

The result is the same.

Conclusion: Reflecting across the x-axis and then across the y-axis has the same effect as a 180^\circ rotation about the origin.

This is an example of describing a sequence of transformations as one single transformation.

7. Worked Example 3: rotation, then translation

Rectangle \(RSTU\) has vertices:

$$R(1,1), \quad S(4,1), \quad T(4,3), \quad U(1,3)$$

First rotate the rectangle \(90^\circ\) counterclockwise about the origin. Then translate it left 2 units.

Step 1: Rotate \(90^\circ\) counterclockwise

The rule is:

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

• \(R(1,1) \rightarrow R'(-1,1)\)
• \(S(4,1) \rightarrow S'(-1,4)\)
• \(T(4,3) \rightarrow T'(-3,4)\)
• \(U(1,3) \rightarrow U'(-3,1)\)

Step 2: Translate left 2 units

The rule is:

$$ (x,y) \rightarrow (x-2,y) $$

• \(R'(-1,1) \rightarrow R''(-3,1)\)
• \(S'(-1,4) \rightarrow S''(-3,4)\)
• \(T'(-3,4) \rightarrow T''(-5,4)\)
• \(U'(-3,1) \rightarrow U''(-5,1)\)

Final image:

$$R''(-3,1), \quad S''(-3,4), \quad T''(-5,4), \quad U''(-5,1)$$

8. Worked Example 4: dilation, then reflection

Triangle \(JKL\) has vertices:

$$J(1,2), \quad K(3,2), \quad L(2,5)$$

Dilate the triangle by a scale factor of \(2\) about the origin, then reflect it across the line \(y=x\).

Step 1: Dilation by scale factor 2

The rule is:

$$ (x,y) \rightarrow (2x,2y) $$

• \(J(1,2) \rightarrow J'(2,4)\)
• \(K(3,2) \rightarrow K'(6,4)\)
• \(L(2,5) \rightarrow L'(4,10)\)

Step 2: Reflect across \(y=x\)

The rule is:

$$ (x,y) \rightarrow (y,x) $$

• \(J'(2,4) \rightarrow J''(4,2)\)
• \(K'(6,4) \rightarrow K''(4,6)\)
• \(L'(4,10) \rightarrow L''(10,4)\)

Final image:

$$J''(4,2), \quad K''(4,6), \quad L''(10,4)$$

What changed here?

• The dilation changed the size of the triangle.
• The reflection changed its orientation.

9. Recognizing equivalent single transformations

Sometimes a sequence of transformations can be replaced by one transformation.

Here are some useful patterns:

• Two translations can be combined into one translation.
• Reflect across the x-axis, then across the y-axis = 180^\circ rotation about the origin.
• Two reflections across the same line bring the figure back to its original position.

Example with translations:

Translate right 4 units, then left 1 unit.

The total movement is right 3 units.

So the single transformation is:

$$ (x,y) \rightarrow (x+3,y) $$

Example with up and down movement:

Translate up 5 units, then down 2 units.

The total movement is up 3 units.

So the single transformation is:

$$ (x,y) \rightarrow (x,y+3) $$

10. Common mistakes to avoid

• Doing the transformations in the wrong order. Always follow the order given.
• Using the original coordinates every time. After the first step, use the new image points.
• Mixing up reflection rules. For example, across the x-axis changes the sign of \(y\), not \(x\).
• Mixing up rotation rules. Check whether it is clockwise or counterclockwise.
• Forgetting that dilation changes size. Multiply both coordinates by the scale factor if the center is the origin.

11. Quick check questions

Try these on your own.

1. Start with \((3,2)\). Translate left 4 units, then reflect across the y-axis.
2. Start with \((1,-5)\). Reflect across the x-axis, then rotate \(180^\circ\) about the origin.
3. Triangle vertices are \((0,1)\), \((2,1)\), and \((1,3)\). Dilate by scale factor \(3\), then translate down 2 units.

Answers

1. \((3,2) \rightarrow (-1,2) \rightarrow (1,2)\)
2. \((1,-5) \rightarrow (1,5) \rightarrow (-1,-5)\)
3. After dilation: \((0,3)\), \((6,3)\), \((3,9)\); after translating down 2: \((0,1)\), \((6,1)\), \((3,7)\)

12. Summary

Successive transformations are transformations done one after another.

To solve problems, apply one step at a time and rewrite the coordinates after each step.

Remember that order matters, and some sequences can be described by a single equivalent transformation.

If you stay organized and use the correct coordinate rules, you can track any figure through multiple transformations with confidence.

Spatial Transformations are ways to move a shape on a flat surface, like paper or a grid, without changing its size or shape.

In 4th Grade, the three main spatial transformations are translations, reflections, and rotations.

These moves help us describe how shapes change position. Even when a shape moves, the sides and angles stay the same.

Important idea: A transformation changes where a figure is or how it faces, but it does not make the figure bigger or smaller.

Let’s learn each kind of transformation.

1. Translation

A translation means a shape slides from one place to another.

When a shape is translated:

• It does not turn.
• It does not flip.
• Its size stays the same.
• Its shape stays the same.

You can think of a translation as a smooth slide.

If a square moves 3 units right and 2 units up, every point on the square moves the same way:

$$3\text{ units right and }2\text{ units up}$$

This means the whole shape slides together.

2. Reflection

A reflection means a shape flips over a line. This line is called the line of reflection.

When a shape is reflected:

• It flips to the other side of a line.
• It keeps the same size and shape.
• It looks like a mirror image.

If the line of reflection is vertical, the shape flips left to right. If the line is horizontal, the shape flips up and down.

A reflection helps us talk about symmetry. If one half of a figure matches the other half across a line, the figure has line symmetry.

3. Rotation

A rotation means a shape turns around a fixed point.

When a shape rotates:

• It turns.
• It does not slide only.
• It does not flip like a reflection.
• Its size and shape stay the same.

A shape can turn a quarter turn, half turn, or full turn.

• Quarter turn = 90°
• Half turn = 180°
• Full turn = 360°

These turns can happen clockwise or counterclockwise.

For example:

$$90^\circ,\ 180^\circ,\ 360^\circ$$

How transformations keep figures the same

After a translation, reflection, or rotation, the figure still has:

• the same side lengths,
• the same angle sizes,
• the same number of sides and corners.

So if a rectangle is moved, reflected, or rotated, it is still a rectangle.

Using position words on a grid

Many transformations are shown on a grid. A grid helps us count how far a shape moves.

On a grid, we can describe moves using words like:

• left
• right
• up
• down

For example, a point might move:

$$4\text{ units left}$$

or

$$2\text{ units down}$$

Each point in the figure must move the same amount in a translation.

Worked Example 1: Translation

A triangle is moved 5 squares to the right.

What kind of transformation is this?

Step 1: Ask what the shape is doing.

The triangle is sliding.

Step 2: Check if it turns or flips.

No, it does not turn or flip.

Answer: This is a translation.

Why? A translation is a slide, and the triangle moved right without changing shape or direction.

Worked Example 2: Reflection

A heart shape is on the left side of a vertical line. Another matching heart shape is on the right side of the line, the same distance away, but reversed like a mirror image.

What transformation happened?

Step 1: Notice the line in the middle.

This is the line of reflection.

Step 2: Notice the shape is flipped across the line.

Answer: This is a reflection.

Why? The shape flipped over a line and became a mirror image.

Worked Example 3: Rotation

An arrow that points up is turned so it points right.

What transformation happened?

Step 1: Ask if the arrow slid, flipped, or turned.

It turned.

Step 2: Find how much it turned.

From up to right is a quarter turn.

Answer: This is a rotation of 90°.

Why? The arrow turned, but it stayed the same size and shape.

Worked Example 4: Choosing the transformation

A square starts in one place on a grid.

Then it appears in a new place:

• same size,
• same direction,
• not flipped,
• 3 units up and 4 units left.

What transformation is it?

Step 1: The square changed place.

Step 2: It did not turn.

Step 3: It did not flip.

Answer: It is a translation.

Why? The square only slid on the grid.

How to tell the transformations apart

• Translation = slide
• Reflection = flip
• Rotation = turn

A good way to remember:

• If it slides, it is a translation.
• If it looks like a mirror image, it is a reflection.
• If it turns around a point, it is a rotation.

Things to watch out for

• A shape that moves to a new place is not always a translation. Check whether it also turns or flips.
• A reflection needs a line of reflection.
• A rotation changes the direction the shape faces.
• None of these transformations change the size of the shape.

Why this matters

Spatial transformations help us understand maps, patterns, art, building designs, and puzzle shapes.

They also help us describe figures using lines, angles, symmetry, and position.

Quick review

1. A translation is a slide.
2. A reflection is a flip over a line.
3. A rotation is a turn around a point.
4. In all three transformations, the figure keeps the same size and shape.

Summary

Spatial transformations are ways to move shapes without changing their size or shape. The three main transformations are translation, reflection, and rotation.

Remember: slide, flip, turn. If you can tell how a shape moved, you can name the transformation.