Chapter Notes: Performing Transformations

Table of Contents
1. Understanding Transformations and the Coordinate Plane
2. Translations
3. Reflections
4. Rotations
5. Dilations
6. Compositions of Transformations
7. Verifying Transformations
8. Common Mistakes to Avoid
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In geometry, we often want to move shapes around, flip them, turn them, or change their size while keeping their basic form. These movements are called transformations. Think of transformations like giving instructions to move a puzzle piece on a board-you might slide it to the left, spin it around, or flip it over. Understanding how to perform transformations helps us analyze symmetry, create designs, and solve real-world problems in fields like architecture, computer graphics, and engineering. In this chapter, you will learn how to perform four fundamental types of transformations: translations, reflections, rotations, and dilations.

Understanding Transformations and the Coordinate Plane

A transformation is a rule that moves or changes a figure in a specific way. The original figure is called the pre-image, and the figure after the transformation is called the image. We often use the coordinate plane to perform transformations because it allows us to describe precisely where each point moves.

When working with transformations, we typically label points on the pre-image with letters like \( A \), \( B \), and \( C \). The corresponding points on the image are labeled with prime notation: \( A' \) (read as "A prime"), \( B' \) (read as "B prime"), and \( C' \) (read as "C prime").

There are two main categories of transformations:

• Rigid transformations (or isometries) preserve both the size and shape of the figure. These include translations, reflections, and rotations.
• Non-rigid transformations change the size of the figure but preserve its shape. The primary example is dilation.

Translations

A translation is a transformation that slides every point of a figure the same distance in the same direction. Imagine sliding a book across a table without turning it or flipping it-that's a translation.

To describe a translation, we specify the horizontal movement (left or right) and the vertical movement (up or down). We can write a translation using coordinate notation:

In this notation, \( a \) represents the horizontal shift (positive means right, negative means left), and \( b \) represents the vertical shift (positive means up, negative means down). Every point \( (x, y) \) in the pre-image moves to a new location \( (x + a, y + b) \) in the image.

Performing a Translation

To translate a figure on the coordinate plane, follow these steps:

1. Identify the coordinates of each vertex of the pre-image.
2. Apply the translation rule to each point by adding the horizontal shift to the x-coordinate and the vertical shift to the y-coordinate.
3. Plot the new points on the coordinate plane.
4. Connect the new points to form the image.

Example:  Triangle ABC has vertices at \( A(2, 3) \), \( B(5, 3) \), and \( C(4, 6) \).
Translate the triangle 4 units left and 2 units down.

What are the coordinates of the image triangle \( A'B'C' \)?

Solution:

The translation rule is \( (x, y) \rightarrow (x - 4, y - 2) \) because moving left means subtracting from x, and moving down means subtracting from y.

For point A:
\( A(2, 3) \rightarrow A'(2 - 4, 3 - 2) = A'(-2, 1) \)

For point B:
\( B(5, 3) \rightarrow B'(5 - 4, 3 - 2) = B'(1, 1) \)

For point C:
\( C(4, 6) \rightarrow C'(4 - 4, 6 - 2) = C'(0, 4) \)

The vertices of the translated triangle are \( A'(-2, 1) \), \( B'(1, 1) \), and \( C'(0, 4) \).

Reflections

A reflection is a transformation that flips a figure over a line, creating a mirror image. The line is called the line of reflection. Think of looking at yourself in a mirror-your reflection appears on the opposite side of the mirror at the same distance.

The most common lines of reflection are the x-axis, the y-axis, and the lines \( y = x \) and \( y = -x \). Each line of reflection has its own rule for transforming coordinates.

Reflection Rules for Common Lines

Performing a Reflection

To reflect a figure over a line:

1. Identify the line of reflection.
2. Apply the appropriate coordinate rule to each vertex of the pre-image.
3. Plot the new points on the coordinate plane.
4. Connect the new points to form the image.

Example:  Quadrilateral DEFG has vertices at \( D(1, 2) \), \( E(3, 5) \), \( F(6, 4) \), and \( G(4, 1) \).
Reflect the quadrilateral over the y-axis.

What are the coordinates of the image \( D'E'F'G' \)?

Solution:

When reflecting over the y-axis, we use the rule \( (x, y) \rightarrow (-x, y) \).

For point D:
\( D(1, 2) \rightarrow D'(-1, 2) \)

For point E:
\( E(3, 5) \rightarrow E'(-3, 5) \)

For point F:
\( F(6, 4) \rightarrow F'(-6, 4) \)

For point G:
\( G(4, 1) \rightarrow G'(-4, 1) \)

The vertices of the reflected quadrilateral are \( D'(-1, 2) \), \( E'(-3, 5) \), \( F'(-6, 4) \), and \( G'(-4, 1) \).

Rotations

A rotation is a transformation that turns a figure around a fixed point called the center of rotation. The amount of turning is measured by the angle of rotation, and the direction can be clockwise or counterclockwise. Imagine spinning a wheel around its center-that's a rotation.

Unless stated otherwise, rotations are performed counterclockwise around the origin \( (0, 0) \). The most common rotation angles are 90°, 180°, and 270°.

Rotation Rules Around the Origin

Performing a Rotation

To rotate a figure around the origin:

1. Identify the angle and direction of rotation.
2. Apply the appropriate coordinate rule to each vertex.
3. Plot the new points on the coordinate plane.
4. Connect the new points to form the image.

Example:  Triangle PQR has vertices at \( P(3, 2) \), \( Q(5, 7) \), and \( R(6, 1) \).
Rotate the triangle 90° counterclockwise around the origin.

What are the coordinates of the image \( P'Q'R' \)?

Solution:

For a 90° counterclockwise rotation, we use the rule \( (x, y) \rightarrow (-y, x) \).

For point P:
\( P(3, 2) \rightarrow P'(-2, 3) \)

For point Q:
\( Q(5, 7) \rightarrow Q'(-7, 5) \)

For point R:
\( R(6, 1) \rightarrow R'(-1, 6) \)

The vertices of the rotated triangle are \( P'(-2, 3) \), \( Q'(-7, 5) \), and \( R'(-1, 6) \).

Rotations Around Points Other Than the Origin

Sometimes we need to rotate a figure around a point that is not the origin. This process is more complex and involves these steps:

1. Translate the entire figure so that the center of rotation moves to the origin.
2. Perform the rotation using the standard rules.
3. Translate the figure back by reversing the initial translation.

Example:  Point A is at \( (5, 3) \).
Rotate point A 180° around the point \( C(2, 1) \).

What are the coordinates of \( A' \)?

Solution:

Step 1: Translate so that C moves to the origin.
The translation is \( (x, y) \rightarrow (x - 2, y - 1) \).
\( A(5, 3) \rightarrow (5 - 2, 3 - 1) = (3, 2) \)

Step 2: Rotate 180° around the origin.
The rule is \( (x, y) \rightarrow (-x, -y) \).
\( (3, 2) \rightarrow (-3, -2) \)

Step 3: Translate back by reversing the initial translation.
The reverse translation is \( (x, y) \rightarrow (x + 2, y + 1) \).
\( (-3, -2) \rightarrow (-3 + 2, -2 + 1) = (-1, -1) \)

The image of point A after rotation is \( A'(-1, -1) \).

Dilations

A dilation is a transformation that changes the size of a figure by a scale factor while keeping its shape the same. The dilation is performed with respect to a fixed point called the center of dilation. Think of using a photocopier to enlarge or reduce an image-the shape stays the same, but the size changes.

The scale factor, denoted by \( k \), determines how the size changes:

• If \( k > 1 \), the image is larger than the pre-image (enlargement).
• If \( 0 < k="">< 1="" \),="" the="" image="" is="" smaller="" than="" the="" pre-image="">
• If \( k = 1 \), the image is the same size as the pre-image (no change).

Dilation from the Origin

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

This means we multiply both coordinates by the scale factor \( k \).

Performing a Dilation

To dilate a figure from the origin:

1. Identify the scale factor \( k \).
2. Multiply the coordinates of each vertex by \( k \).
3. Plot the new points on the coordinate plane.
4. Connect the new points to form the image.

Example:  Rectangle JKLM has vertices at \( J(2, 4) \), \( K(6, 4) \), \( L(6, 2) \), and \( M(2, 2) \).
Dilate the rectangle from the origin with a scale factor of \( \frac{1}{2} \).

What are the coordinates of the image \( J'K'L'M' \)?

Solution:

The dilation rule is \( (x, y) \rightarrow (\frac{1}{2}x, \frac{1}{2}y) \).

For point J:
\( J(2, 4) \rightarrow J'(\frac{1}{2} \times 2, \frac{1}{2} \times 4) = J'(1, 2) \)

For point K:
\( K(6, 4) \rightarrow K'(\frac{1}{2} \times 6, \frac{1}{2} \times 4) = K'(3, 2) \)

For point L:
\( L(6, 2) \rightarrow L'(\frac{1}{2} \times 6, \frac{1}{2} \times 2) = L'(3, 1) \)

For point M:
\( M(2, 2) \rightarrow M'(\frac{1}{2} \times 2, \frac{1}{2} \times 2) = M'(1, 1) \)

The vertices of the dilated rectangle are \( J'(1, 2) \), \( K'(3, 2) \), \( L'(3, 1) \), and \( M'(1, 1) \).

Properties of Dilations

Dilations have several important properties:

• The image is similar to the pre-image (same shape, different size).
• Corresponding angles remain congruent.
• The lengths of corresponding sides are proportional, with the ratio equal to the scale factor.
• The center of dilation is the only point that remains fixed (maps to itself).
• Lines connecting corresponding points on the pre-image and image all pass through the center of dilation.

Compositions of Transformations

A composition of transformations occurs when we perform two or more transformations in sequence. The order in which transformations are performed can affect the final result, especially when the transformations are not all of the same type.

To perform a composition:

1. Apply the first transformation to the pre-image to get an intermediate image.
2. Apply the second transformation to the intermediate image to get the final image.
3. Continue with additional transformations if needed.

Example:  Point S is at \( (4, 1) \).
First, reflect point S over the x-axis.
Then, translate the result 3 units left and 2 units up.

What are the coordinates of the final image \( S'' \)?

Solution:

Step 1: Reflect over the x-axis using \( (x, y) \rightarrow (x, -y) \).
\( S(4, 1) \rightarrow S'(4, -1) \)

Step 2: Translate 3 units left and 2 units up using \( (x, y) \rightarrow (x - 3, y + 2) \).
\( S'(4, -1) \rightarrow S''(4 - 3, -1 + 2) = S''(1, 1) \)

The final coordinates of point S after both transformations are \( S''(1, 1) \).

Verifying Transformations

After performing a transformation, it's important to verify that you've applied the rule correctly. Here are some strategies:

• For translations: Check that every point moved the same distance in the same direction. Measure the horizontal and vertical distances.
• For reflections: Verify that corresponding points are equidistant from the line of reflection and that the line of reflection is the perpendicular bisector of segments connecting corresponding points.
• For rotations: Confirm that the distance from each point to the center of rotation remains the same. Check that the angle formed is correct.
• For dilations: Calculate the distances from the center of dilation to corresponding points and verify that the ratio equals the scale factor.

Common Mistakes to Avoid

When performing transformations, watch out for these common errors:

• Confusing the signs when reflecting (forgetting which coordinate changes sign).
• Mixing up the order of coordinates when rotating (especially for 90° rotations).
• Applying the wrong direction for rotations (clockwise vs. counterclockwise).
• Forgetting to apply the transformation to all vertices of a figure.
• In compositions, applying transformations in the wrong order.
• Multiplying by the scale factor incorrectly or only applying it to one coordinate.

By understanding and practicing these four fundamental transformations-translations, reflections, rotations, and dilations-you build a strong foundation for studying geometry, symmetry, and coordinate geometry. Transformations help us see relationships between figures and understand how shapes can be manipulated while preserving important properties. Whether you're creating computer graphics, analyzing architectural plans, or solving geometric proofs, the ability to perform transformations accurately is an essential mathematical skill.