Unit 9 Transformations Homework 3 Rotations Answer Key

Let's delve into the fascinating world of rotations, a fundamental type of transformation in geometry. Understanding rotations is crucial not just for acing your math homework, but also for grasping concepts in computer graphics, physics, and even art. This article will provide a comprehensive guide to rotations, complete with examples and solutions to help you master this topic.

Understanding Rotations: The Basics

A rotation is a transformation that turns a figure about a fixed point, known as the center of rotation. Imagine pinning a piece of paper to a board and rotating the paper around the pin – that's essentially what a rotation does in geometry.

Here are the key elements of a rotation:

• Center of Rotation: This is the point around which the figure is rotated. Think of it as the anchor point.
• Angle of Rotation: This specifies the amount of rotation, usually measured in degrees. A positive angle indicates a counterclockwise rotation, while a negative angle indicates a clockwise rotation.
• Direction of Rotation: As mentioned above, this can be either clockwise or counterclockwise.

Coordinate Plane Rotations

Rotations are commonly performed on the coordinate plane, where points are defined by their x and y coordinates. When rotating a point around the origin (0, 0), specific rules apply that change the coordinates of the point. These rules are based on trigonometric functions like sine and cosine, but we'll focus on the practical applications for common angles like 90°, 180°, and 270°.

Rotation Rules Around the Origin

Here's a handy summary of the rotation rules around the origin in the coordinate plane:

• 90° Counterclockwise Rotation: (x, y) → (-y, x)
• 180° Rotation: (x, y) → (-x, -y)
• 270° Counterclockwise Rotation: (x, y) → (y, -x)
• 360° Rotation: (x, y) → (x, y) (Returns to the original point)

It's crucial to memorize these rules or have them readily available when solving rotation problems. Let's illustrate these rules with examples.

Examples of Rotations

Example 1: Rotating a Point 90° Counterclockwise

Let's rotate the point (2, 3) by 90° counterclockwise around the origin. Using the rule (x, y) → (-y, x), we get:

(2, 3) → (-3, 2)

So, the image of the point (2, 3) after a 90° counterclockwise rotation is (-3, 2).

Example 2: Rotating a Point 180°

Rotate the point (-1, 4) by 180° around the origin. Using the rule (x, y) → (-x, -y), we get:

(-1, 4) → (1, -4)

The image of the point (-1, 4) after a 180° rotation is (1, -4).

Example 3: Rotating a Point 270° Counterclockwise

Rotate the point (5, -2) by 270° counterclockwise around the origin. Using the rule (x, y) → (y, -x), we get:

(5, -2) → (-2, -5)

The image of the point (5, -2) after a 270° counterclockwise rotation is (-2, -5).

Solving Rotation Problems: Step-by-Step

Now, let's tackle some more complex rotation problems. The key is to break down the problem into smaller, manageable steps.

Problem 1: Rotating a Triangle

Triangle ABC has vertices A(1, 1), B(4, 1), and C(4, 3). Rotate triangle ABC 90° counterclockwise around the origin. Find the coordinates of the image triangle A'B'C'.

Solution:

1. Apply the 90° rotation rule to each vertex:

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

2. The coordinates of the image triangle are: A'(-1, 1), B'(-1, 4), and C'(-3, 4).

Problem 2: Rotating a Quadrilateral

Quadrilateral PQRS has vertices P(-2, -1), Q(-1, 2), R(2, 2), and S(1, -1). Rotate quadrilateral PQRS 180° around the origin. Find the coordinates of the image quadrilateral P'Q'R'S'.

Solution:

1. Apply the 180° rotation rule to each vertex:

   • P(-2, -1) → P'(2, 1)
   • Q(-1, 2) → Q'(1, -2)
   • R(2, 2) → R'(-2, -2)
   • S(1, -1) → S'(-1, 1)

2. The coordinates of the image quadrilateral are: P'(2, 1), Q'(1, -2), R'(-2, -2), and S'(-1, 1).

Problem 3: Rotating a Line Segment

Line segment DE has endpoints D(-3, 0) and E(0, 4). Rotate line segment DE 270° counterclockwise around the origin. Find the coordinates of the image line segment D'E'.

Solution:

1. Apply the 270° rotation rule to each endpoint:

   • D(-3, 0) → D'(0, 3)
   • E(0, 4) → E'(4, 0)

2. The coordinates of the image line segment are: D'(0, 3) and E'(4, 0).

Beyond the Origin: Rotations Around Other Points

While rotations around the origin are the most common type encountered in introductory geometry, it's important to understand that rotations can occur around any point in the plane.

To rotate a figure around a point other than the origin, you'll need to perform a series of transformations:

1. Translation: Translate the figure so that the center of rotation coincides with the origin.
2. Rotation: Rotate the translated figure around the origin using the standard rotation rules.
3. Translation (Reverse): Translate the rotated figure back to its original position by applying the inverse of the initial translation.

This process can be mathematically complex, but it allows for rotations around any arbitrary point.

Unit 9 Transformations Homework 3 Rotations: Sample Problems and Solutions

Now, let's address the specific topic of "Unit 9 Transformations Homework 3 Rotations" by providing sample problems and their solutions. These problems are designed to reflect the types of questions you might encounter in such an assignment.

Problem 1:

Triangle JKL has vertices J(2, -1), K(4, -1), and L(4, 2). Rotate triangle JKL 90° counterclockwise around the origin.

a) What are the coordinates of J'K'L'?

b) Graph triangle JKL and its image J'K'L'.

Solution:

a) Applying the 90° counterclockwise rotation rule (x, y) → (-y, x):

• J(2, -1) → J'(1, 2)
• K(4, -1) → K'(1, 4)
• L(4, 2) → L'(-2, 4)

The coordinates of J'K'L' are: J'(1, 2), K'(1, 4), and L'(-2, 4).

b) To graph the triangles, plot the original points J(2, -1), K(4, -1), and L(4, 2), and connect them to form triangle JKL. Then, plot the image points J'(1, 2), K'(1, 4), and L'(-2, 4), and connect them to form triangle J'K'L'. You'll observe that triangle J'K'L' is a 90° counterclockwise rotation of triangle JKL around the origin.

Problem 2:

Rectangle ABCD has vertices A(-3, 1), B(-1, 1), C(-1, 3), and D(-3, 3). Rotate rectangle ABCD 180° around the origin.

a) What are the coordinates of A'B'C'D'?

b) Describe the relationship between rectangle ABCD and rectangle A'B'C'D'.

Solution:

a) Applying the 180° rotation rule (x, y) → (-x, -y):

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

The coordinates of A'B'C'D' are: A'(3, -1), B'(1, -1), C'(1, -3), and D'(3, -3).

b) Rectangle A'B'C'D' is a 180° rotation of rectangle ABCD around the origin. This means that the two rectangles are congruent (identical in size and shape) but have opposite orientations. They are symmetrical with respect to the origin.

Problem 3:

Point P has coordinates (5, -2). Rotate point P 270° counterclockwise around the origin. What are the coordinates of P'?

Solution:

Applying the 270° counterclockwise rotation rule (x, y) → (y, -x):

• P(5, -2) → P'(-2, -5)

The coordinates of P' are (-2, -5).

Problem 4:

A figure is rotated 90 degrees clockwise about the origin. The image of point A is A'(-4,6). What were the original coordinates of point A?

Solution:

A 90 degree clockwise rotation is the same as a 270 degree counter-clockwise rotation. The rule for a 270 degree counter-clockwise rotation is (x, y) -> (y, -x). To reverse this, we need to apply the inverse. If A' is (-4, 6), that means that after the rotation, y = -4 and -x = 6. Solving for x, we get x = -6. Therefore, the original point A was (-6, -4).

Problem 5:

Triangle FGH has vertices F(-2, -2), G(-1, 0), and H(0, -2). Rotate triangle FGH 90° clockwise around the origin.

a) What are the coordinates of F'G'H'?

Solution:

A 90 degree clockwise rotation is the same as a 270 degree counter-clockwise rotation. Applying the 270° counterclockwise rotation rule (x, y) → (y, -x):

• F(-2, -2) → F'(-2, 2)
• G(-1, 0) → G'(0, 1)
• H(0, -2) → H'(-2, 0)

The coordinates of F'G'H' are: F'(-2, 2), G'(0, 1), and H'(-2, 0).

Tips for Success with Rotation Problems

Here are some helpful tips to keep in mind when working with rotation problems:

• Memorize the rotation rules: Knowing the rules for 90°, 180°, and 270° rotations around the origin is essential.
• Practice, practice, practice: The more you practice, the more comfortable you'll become with applying the rotation rules.
• Draw diagrams: Visualizing the rotation can help you understand the transformation and avoid mistakes.
• Pay attention to the direction of rotation: Be careful to distinguish between clockwise and counterclockwise rotations.
• Double-check your work: After applying a rotation, make sure the image coordinates make sense in relation to the original coordinates.

Real-World Applications of Rotations

Rotations aren't just abstract mathematical concepts; they have numerous real-world applications:

• Computer Graphics: Rotations are fundamental in computer graphics for creating 3D models, animations, and games.
• Physics: Rotational motion is a key concept in physics, describing the movement of objects around an axis.
• Engineering: Engineers use rotations in designing machines, structures, and other systems.
• Navigation: Rotations are used in navigation systems to determine the orientation of vehicles and aircraft.
• Art and Design: Artists and designers use rotations to create symmetrical patterns, repeating designs, and visually appealing compositions.

Conclusion

Mastering rotations is a crucial step in your journey through geometry and beyond. By understanding the basic principles, memorizing the rotation rules, and practicing with examples, you'll be well-equipped to tackle any rotation problem that comes your way. Remember to break down complex problems into smaller steps, visualize the transformations, and double-check your work. With dedication and practice, you can unlock the power of rotations and apply them to a wide range of real-world applications. Good luck with your "Unit 9 Transformations Homework 3 Rotations"! You've got this!