Unit 9 Test Study Guide Transformations

Transformations in geometry unlock a world of understanding how shapes can be manipulated while retaining certain properties. A unit test focusing on transformations requires a solid grasp of translations, reflections, rotations, and dilations, along with their properties and how they interact. This study guide will delve into each type of transformation, providing explanations, examples, and practice problems to ensure you’re well-prepared for your test.

Understanding Geometric Transformations

Geometric transformations involve changing the position, size, or orientation of a shape or figure. The original shape is known as the preimage, and the new shape after the transformation is called the image. Transformations can be categorized as:

• Translations: Sliding a shape without rotating or resizing it.
• Reflections: Flipping a shape over a line.
• Rotations: Turning a shape around a point.
• Dilations: Resizing a shape by a scale factor.

Each of these transformations has specific rules and properties that determine how the coordinates of a point on the preimage change to those of the image.

1. Translations: Sliding Shapes

A translation moves every point of a figure the same distance in the same direction. This movement is defined by a translation vector, which specifies the horizontal and vertical shift.

How Translations Work

A translation is defined by the rule:

(x, y) → (x + a, y + b)

Where:

• (x, y) are the coordinates of a point on the preimage.
• (a, b) is the translation vector.
• (x + a, y + b) are the coordinates of the corresponding point on the image.

Properties of Translations

• Congruence: Translations preserve the size and shape of the figure. The preimage and image are congruent.
• Orientation: Translations preserve the orientation of the figure. The image faces the same way as the preimage.
• Distance: The distance between any two points on the preimage is the same as the distance between their corresponding points on the image.
• Parallelism: Parallel lines remain parallel after a translation.

Examples of Translations

Example 1:

Translate triangle ABC with vertices A(1, 2), B(3, 4), and C(5, 1) using the translation vector (2, -3).

Solution:

Apply the translation rule (x, y) → (x + 2, y - 3) to each vertex:

• A(1, 2) → A'(1 + 2, 2 - 3) = A'(3, -1)
• B(3, 4) → B'(3 + 2, 4 - 3) = B'(5, 1)
• C(5, 1) → C'(5 + 2, 1 - 3) = C'(7, -2)

The vertices of the translated triangle A'B'C' are A'(3, -1), B'(5, 1), and C'(7, -2).

Example 2:

A point P(4, -1) is translated to P'(-2, 3). Find the translation vector.

Solution:

Let the translation vector be (a, b). Then:

• 4 + a = -2 => a = -6
• -1 + b = 3 => b = 4

The translation vector is (-6, 4).

Practice Problems for Translations

1. Translate the square with vertices P(0, 0), Q(2, 0), R(2, 2), and S(0, 2) using the translation vector (-1, 3). Find the coordinates of the image.
2. Triangle XYZ has vertices X(-3, 1), Y(-1, 4), and Z(2, 2). If the triangle is translated by (5, -2), find the coordinates of the translated triangle X'Y'Z'.
3. A line segment AB with endpoints A(2, -5) and B(6, -2) is translated such that A' is at (0, -1). Find the translation vector and the coordinates of B'.

2. Reflections: Flipping Shapes

A reflection flips a figure over a line, known as the line of reflection. The image is a mirror image of the preimage.

How Reflections Work

Reflections can occur over various lines, but the most common are the x-axis, y-axis, and the lines y = x and y = -x.

• Reflection over the x-axis: (x, y) → (x, -y)
• Reflection over the y-axis: (x, y) → (-x, y)
• Reflection over the line y = x: (x, y) → (y, x)
• Reflection over the line y = -x: (x, y) → (-y, -x)

Properties of Reflections

• Congruence: Reflections preserve the size and shape of the figure. The preimage and image are congruent.
• Orientation: Reflections reverse the orientation of the figure. If you go clockwise around the vertices of the preimage, you will go counterclockwise around the vertices of the image.
• Distance: The distance from any point on the preimage to the line of reflection is the same as the distance from its corresponding point on the image to the line of reflection.
• Perpendicularity: The line connecting a point on the preimage to its corresponding point on the image is perpendicular to the line of reflection.

Examples of Reflections

Example 1:

Reflect triangle PQR with vertices P(1, 2), Q(3, 4), and R(5, 1) over the x-axis.

Solution:

Apply the reflection rule (x, y) → (x, -y) to each vertex:

• P(1, 2) → P'(1, -2)
• Q(3, 4) → Q'(3, -4)
• R(5, 1) → R'(5, -1)

The vertices of the reflected triangle P'Q'R' are P'(1, -2), Q'(3, -4), and R'(5, -1).

Example 2:

Reflect the point A(-2, 5) over the line y = x.

Solution:

Apply the reflection rule (x, y) → (y, x):

• A(-2, 5) → A'(5, -2)

The reflected point A' is (5, -2).

Example 3:

Reflect the square ABCD with vertices A(1, 1), B(3, 1), C(3, 3), and D(1, 3) over the y-axis.

Solution:

Apply the reflection rule (x, y) → (-x, y) to each vertex:

• A(1, 1) → A'(-1, 1)
• B(3, 1) → B'(-3, 1)
• C(3, 3) → C'(-3, 3)
• D(1, 3) → D'(-1, 3)

The vertices of the reflected square A'B'C'D' are A'(-1, 1), B'(-3, 1), C'(-3, 3), and D'(-1, 3).

Practice Problems for Reflections

1. Reflect the triangle with vertices A(2, 3), B(4, 1), and C(1, 1) over the y-axis. Find the coordinates of the image.
2. Reflect the rectangle with vertices P(-2, -1), Q(2, -1), R(2, 3), and S(-2, 3) over the x-axis. Find the coordinates of the image.
3. Reflect the point (5, -3) over the line y = x.
4. Reflect the line segment joining points A(1, 4) and B(5, 2) over the line y = -x.

3. Rotations: Turning Shapes

A rotation turns a figure around a fixed point, known as the center of rotation. The amount of rotation is measured in degrees.

How Rotations Work

Rotations are typically defined by the angle of rotation and the center of rotation. The most common center of rotation is the origin (0, 0).

• Rotation of 90° counterclockwise about the origin: (x, y) → (-y, x)
• Rotation of 180° about the origin: (x, y) → (-x, -y)
• Rotation of 270° counterclockwise about the origin: (x, y) → (y, -x)
• Rotation of 90° clockwise about the origin: (x, y) → (y, -x)
• Rotation of 270° clockwise about the origin: (x, y) → (-y, x)

Properties of Rotations

• Congruence: Rotations preserve the size and shape of the figure. The preimage and image are congruent.
• Orientation: Rotations preserve the orientation of the figure relative to the center of rotation.
• Distance: The distance from any point on the preimage to the center of rotation is the same as the distance from its corresponding point on the image to the center of rotation.
• Angle Measure: The angle between any two lines on the preimage is the same as the angle between their corresponding lines on the image.

Examples of Rotations

Example 1:

Rotate triangle ABC with vertices A(1, 2), B(3, 4), and C(5, 1) by 90° counterclockwise about the origin.

Solution:

Apply the rotation rule (x, y) → (-y, x) to each vertex:

• A(1, 2) → A'(-2, 1)
• B(3, 4) → B'(-4, 3)
• C(5, 1) → C'(-1, 5)

The vertices of the rotated triangle A'B'C' are A'(-2, 1), B'(-4, 3), and C'(-1, 5).

Example 2:

Rotate the point P(-3, 2) by 180° about the origin.

Solution:

Apply the rotation rule (x, y) → (-x, -y):

• P(-3, 2) → P'(3, -2)

The rotated point P' is (3, -2).

Example 3:

Rotate square DEFG with vertices D(1, -1), E(3, -1), F(3, -3), and G(1, -3) by 270° counterclockwise about the origin.

Solution:

Apply the rotation rule (x, y) → (y, -x) to each vertex:

• D(1, -1) → D'(-1, -1)
• E(3, -1) → E'(-1, -3)
• F(3, -3) → F'(-3, -3)
• G(1, -3) → G'(-3, -1)

The vertices of the rotated square D'E'F'G' are D'(-1, -1), E'(-1, -3), F'(-3, -3), and G'(-3, -1).

Practice Problems for Rotations

1. Rotate the triangle with vertices A(2, 3), B(4, 1), and C(1, 1) by 90° clockwise about the origin. Find the coordinates of the image.
2. Rotate the rectangle with vertices P(-2, -1), Q(2, -1), R(2, 3), and S(-2, 3) by 180° about the origin. Find the coordinates of the image.
3. Rotate the point (5, -3) by 270° clockwise about the origin.
4. Rotate the line segment joining points A(1, 4) and B(5, 2) by 90° counterclockwise about the origin.

4. Dilations: Resizing Shapes

A dilation changes the size of a figure by a scale factor. The figure either becomes larger (enlargement) or smaller (reduction), depending on the scale factor.

How Dilations Work

A dilation is defined by the rule:

(x, y) → (kx, ky)

Where:

• (x, y) are the coordinates of a point on the preimage.
• k is the scale factor.
• (kx, ky) are the coordinates of the corresponding point on the image.

If k > 1, the dilation is an enlargement. If 0 < k < 1, the dilation is a reduction. If k = 1, the dilation results in the same figure.

Properties of Dilations

• Similarity: Dilations preserve the shape of the figure but not necessarily the size. The preimage and image are similar.
• Orientation: Dilations preserve the orientation of the figure. The image faces the same way as the preimage.
• Angle Measure: The angle between any two lines on the preimage is the same as the angle between their corresponding lines on the image.
• Parallelism: Parallel lines remain parallel after a dilation.

Examples of Dilations

Example 1:

Dilate triangle ABC with vertices A(1, 2), B(3, 4), and C(5, 1) by a scale factor of 2.

Solution:

Apply the dilation rule (x, y) → (2x, 2y) to each vertex:

• A(1, 2) → A'(2(1), 2(2)) = A'(2, 4)
• B(3, 4) → B'(2(3), 2(4)) = B'(6, 8)
• C(5, 1) → C'(2(5), 2(1)) = C'(10, 2)

The vertices of the dilated triangle A'B'C' are A'(2, 4), B'(6, 8), and C'(10, 2).

Example 2:

Dilate the point P(4, -6) by a scale factor of 0.5.

Solution:

Apply the dilation rule (x, y) → (0.5x, 0.5y):

• P(4, -6) → P'(0.5(4), 0.5(-6)) = P'(2, -3)

The dilated point P' is (2, -3).

Example 3:

Dilate square ABCD with vertices A(1, 1), B(3, 1), C(3, 3), and D(1, 3) by a scale factor of 3.

Solution:

Apply the dilation rule (x, y) → (3x, 3y) to each vertex:

• A(1, 1) → A'(3(1), 3(1)) = A'(3, 3)
• B(3, 1) → B'(3(3), 3(1)) = B'(9, 3)
• C(3, 3) → C'(3(3), 3(3)) = C'(9, 9)
• D(1, 3) → D'(3(1), 3(3)) = D'(3, 9)

The vertices of the dilated square A'B'C'D' are A'(3, 3), B'(9, 3), C'(9, 9), and D'(3, 9).

Practice Problems for Dilations

1. Dilate the triangle with vertices A(2, 3), B(4, 1), and C(1, 1) by a scale factor of 4. Find the coordinates of the image.
2. Dilate the rectangle with vertices P(-2, -1), Q(2, -1), R(2, 3), and S(-2, 3) by a scale factor of 1.5. Find the coordinates of the image.
3. Dilate the point (5, -3) by a scale factor of 0.25.
4. Dilate the line segment joining points A(1, 4) and B(5, 2) by a scale factor of 2.5.

Combining Transformations

In some problems, you may need to perform multiple transformations in a sequence. The order of transformations matters, as different orders can result in different final images.

Example of Combined Transformations

Example:

Triangle ABC has vertices A(1, 2), B(3, 4), and C(5, 1). First, reflect it over the x-axis, and then translate it by the vector (2, -1). Find the final coordinates of the vertices.

Solution:

1. Reflection over the x-axis:
   • A(1, 2) → A'(1, -2)
   • B(3, 4) → B'(3, -4)
   • C(5, 1) → C'(5, -1)
2. Translation by (2, -1):
   • A'(1, -2) → A''(1 + 2, -2 - 1) = A''(3, -3)
   • B'(3, -4) → B''(3 + 2, -4 - 1) = B''(5, -5)
   • C'(5, -1) → C''(5 + 2, -1 - 1) = C''(7, -2)

The final coordinates of the vertices after both transformations are A''(3, -3), B''(5, -5), and C''(7, -2).

Practice Problems for Combined Transformations

1. Triangle PQR has vertices P(2, 1), Q(4, 3), and R(1, 4). Rotate the triangle 90° counterclockwise about the origin, and then reflect it over the y-axis. Find the final coordinates.
2. Square ABCD has vertices A(-1, -1), B(1, -1), C(1, 1), and D(-1, 1). Dilate the square by a scale factor of 2, and then translate it by the vector (3, 2). Find the final coordinates.
3. Line segment EF has endpoints E(3, -2) and F(5, 1). Reflect the line segment over the line y = x, and then rotate it 180° about the origin. Find the final coordinates.

Identifying Transformations

Sometimes, you will be given a preimage and an image and asked to identify the transformation(s) that occurred. This requires careful observation and knowledge of the properties of each transformation.

Tips for Identifying Transformations

1. Check for Size Changes: If the size changes, it's a dilation.
2. Check for Orientation Changes: If the orientation is reversed, it's a reflection.
3. Look for Sliding: If the figure has simply moved without changing size or orientation, it's a translation.
4. Look for Turning: If the figure has turned around a point, it's a rotation.
5. Consider Multiple Transformations: Sometimes, more than one transformation has occurred.

Examples of Identifying Transformations

Example 1:

Preimage: A(1, 1), B(3, 1), C(3, 3), D(1, 3) Image: A'(3, 3), B'(5, 3), C'(5, 5), D'(3, 5)

Solution:

The size of the figure has changed. The coordinates have been shifted, but the shape remains the same. This is a dilation centered at the origin. To find the scale factor, compare the coordinates of corresponding points: A'(3, 3) / A(1, 1) = 3. So, the scale factor is 3. The transformation is a dilation with a scale factor of 3.

Example 2:

Preimage: P(2, -1), Q(4, -1), R(4, -3), S(2, -3) Image: P'(-2, -1), Q'(-4, -1), R'(-4, -3), S'(-2, -3)

Solution:

The size has not changed, but the x-coordinates have changed signs while the y-coordinates remain the same. This indicates a reflection over the y-axis. The transformation is a reflection over the y-axis.

Example 3:

Preimage: E(1, 2), F(3, 2), G(3, 4), H(1, 4) Image: E'(4, -1), F'(4, -3), G'(2, -3), H'(2, -1)

Solution:

The size is the same, but the figure appears to have turned. The rule (x, y) → (y, -x) maps to the image. This indicates a rotation of 90° clockwise about the origin. The transformation is a rotation of 90° clockwise about the origin.

Advanced Topics in Transformations

Transformations and Matrices

Matrices can be used to represent transformations in a more compact and efficient way. A transformation matrix is a matrix that, when multiplied by the coordinate matrix of a point, gives the coordinates of the transformed point.

For example, the matrix for a 90° counterclockwise rotation about the origin is:

[ 0  -1 ]
[ 1   0 ]

To rotate a point (x, y) by 90° counterclockwise, you would multiply the matrix by the column vector [x, y]:

[ 0  -1 ] [ x ] = [ -y ]
[ 1   0 ] [ y ] = [  x ]

This confirms the transformation rule (x, y) → (-y, x).

Invariant Points

An invariant point is a point that remains unchanged after a transformation. For example, the origin (0, 0) is an invariant point for rotations and dilations centered at the origin. Understanding invariant points can help in analyzing and predicting the behavior of transformations.

Symmetry

Transformations are closely related to the concept of symmetry. A figure has symmetry if it can be transformed onto itself by a transformation. For example, a figure has rotational symmetry if it can be rotated by a certain angle and still look the same. A figure has reflectional symmetry if it can be reflected over a line and still look the same.

Conclusion

Mastering geometric transformations requires a solid understanding of translations, reflections, rotations, and dilations, as well as their properties and how they can be combined. By studying the definitions, properties, examples, and practice problems in this guide, you will be well-prepared for your unit test on transformations. Remember to pay close attention to the order of transformations and to practice identifying transformations from given preimages and images. Good luck with your test!