Lesson 4 Extra Practice Dilations

Lesson 4 Extra Practice: Mastering Dilations

Understanding dilations is crucial for mastering geometry and building a strong foundation in higher-level mathematics. This comprehensive guide provides extra practice problems, detailed explanations, and helpful tips to solidify your understanding of dilations. We'll cover everything from the basics of dilation to more complex applications, ensuring you're well-prepared to tackle any dilation problem. Whether you're struggling with the concept or aiming to perfect your skills, this resource will help you master dilations.

What is a Dilation?

A dilation is a transformation that changes the size of a figure, but not its shape. It's like zooming in or out on a picture. The original figure is called the pre-image, and the new, transformed figure is called the image. Dilations are defined by a center of dilation and a scale factor.

Center of Dilation: This is a fixed point that all points on the pre-image are transformed relative to. Think of it as the "pivot point" for the dilation.
Scale Factor: This is a number that determines the size change. A scale factor greater than 1 enlarges the figure (an enlargement), while a scale factor between 0 and 1 reduces the figure (a reduction). A scale factor of 1 results in a congruent figure (no change in size).

Understanding Scale Factor and Center of Dilation

Let's illustrate with an example. Imagine a triangle with vertices A(1,1), B(3,1), and C(2,3). If we dilate this triangle with a center of dilation at the origin (0,0) and a scale factor of 2, each coordinate will be multiplied by 2.

A(1,1) becomes A'(2,2)
B(3,1) becomes B'(6,2)
C(2,3) becomes C'(4,6)

Notice that the new triangle A'B'C' is similar to the original triangle ABC, but twice as large. The ratios of corresponding side lengths are all equal to the scale factor (2). The angles remain unchanged.

If we used a scale factor of 0.5, the new coordinates would be A'(0.5, 0.5), B'(1.5, 0.5), and C'(1, 1.5), resulting in a smaller, similar triangle.

Step-by-Step Guide to Performing Dilations

Here's a step-by-step guide to help you perform dilations:

Identify the Center of Dilation: Locate the point designated as the center of dilation. This point remains fixed during the transformation.
Determine the Scale Factor: Identify the scale factor (k). This number dictates the enlargement or reduction.
Connect Points to the Center: Draw lines from the center of dilation to each vertex of the pre-image.
Apply the Scale Factor: For each vertex, measure the distance from the center of dilation to the vertex. Multiply this distance by the scale factor. Along the same line, locate the new vertex at the calculated distance from the center of dilation.
Connect the New Vertices: Connect the newly located vertices to form the image.

Practice Problems: Basic Dilations

Let's work through some practice problems to reinforce your understanding. Remember to carefully identify the center of dilation and scale factor before starting.

Problem 1: Dilate a rectangle with vertices A(1,2), B(4,2), C(4,5), and D(1,5) using the origin (0,0) as the center of dilation and a scale factor of 3. Find the coordinates of the image vertices.

Problem 2: A triangle has vertices P(2,1), Q(4,3), and R(6,1). Dilate this triangle with a center of dilation at (1,1) and a scale factor of 1/2. What are the coordinates of the image vertices?

Problem 3: A square with vertices at (1,1), (4,1), (4,4), and (1,4) is dilated with a center at (0,0) and a scale factor of 0.5. Draw the pre-image and image on a coordinate plane and describe the relationship between the two figures.

Solutions and Explanations:

Problem 1 Solution:

A(1,2) becomes A'(3,6) (13 = 3, 23 = 6)
B(4,2) becomes B'(12,6)
C(4,5) becomes C'(12,15)
D(1,5) becomes D'(3,15)

Problem 2 Solution:

This problem involves dilating about a point other than the origin. You need to find the difference between each coordinate of the pre-image points and the center of dilation, multiply that difference by the scale factor, and then add the center of dilation coordinates back.

For P(2,1) the differences are (2-1, 1-1) = (1,0). Multiplying by 1/2 gives (0.5, 0). Adding back (1,1) results in P'(1.5, 1).
For Q(4,3) the differences are (4-1, 3-1) = (3,2). Multiplying by 1/2 gives (1.5, 1). Adding back (1,1) results in Q'(2.5, 2).
For R(6,1) the differences are (6-1, 1-1) = (5,0). Multiplying by 1/2 gives (2.5, 0). Adding back (1,1) results in R'(3.5, 1).

Problem 3 Solution: The resulting image will be a smaller square, similar to the original but with side lengths half the size. The center of the square remains at the origin.

Dilations with Negative Scale Factors

Dilations can also have negative scale factors. A negative scale factor implies a dilation combined with a reflection across the center of dilation. The image will be the same size as calculated by the absolute value of the scale factor, but it will be reflected through the center of dilation.

Example: Dilating a point (2,3) with a center at (0,0) and a scale factor of -2 will result in the point (-4,-6).

Dilations in Real-World Applications

Dilations are not just theoretical concepts. They have numerous real-world applications, including:

Photography: Zooming in or out on a camera is a form of dilation.
Mapmaking: Maps use dilations to represent large areas on a smaller scale.
Engineering: Architects and engineers use dilations to scale up or down designs.
Computer Graphics: Image scaling and resizing in software relies heavily on dilations.

Frequently Asked Questions (FAQ)

Q: What happens if the scale factor is 0? A: If the scale factor is 0, the image will be a single point located at the center of dilation.
Q: Can the center of dilation be outside the pre-image? A: Yes, the center of dilation can be anywhere on the coordinate plane. The transformation still applies the same principles of proportional distances.
Q: What if the pre-image is a curved figure? A: Dilating a curved figure follows the same principles. Each point on the curve is dilated relative to the center and scale factor. The resulting image will be a similar curve.

Advanced Practice Problems

Let's move on to some more challenging problems that incorporate different aspects of dilations.

Problem 4: A circle with a radius of 5 cm is dilated with a scale factor of 3. What is the radius of the image?

Problem 5: Two similar triangles have a scale factor of 2:5. If the perimeter of the smaller triangle is 12 cm, what is the perimeter of the larger triangle?

Problem 6: Dilate the line segment with endpoints A(-2, 4) and B(2, 6) using the origin as the center of dilation and a scale factor of -1.5. Plot both the pre-image and image and describe the transformation.

Solutions and Explanations:

Problem 4 Solution: The radius of the image is 3 * 5 cm = 15 cm. The scale factor applies directly to the linear dimensions of the circle.

Problem 5 Solution: The ratio of perimeters is the same as the scale factor. Therefore, the perimeter of the larger triangle is (5/2) * 12 cm = 30 cm.

Problem 6 Solution:

A(-2, 4) becomes A'(3, -6)
B(2, 6) becomes B'(-3, -9) The transformation involves both a dilation (with a scale factor of 1.5) and a reflection across the origin due to the negative scale factor.

Conclusion

Understanding dilations is a fundamental skill in geometry and related fields. By mastering the concepts of scale factor, center of dilation, and their applications to different shapes and situations, you'll gain a deeper understanding of geometric transformations and their practical uses. Remember to practice consistently, utilizing the step-by-step guide and the various practice problems provided to ensure you develop a strong grasp of this vital topic. Remember to review the concepts covered and tackle additional problems from your textbook or other resources for comprehensive learning. Through diligent practice, you will confidently master the topic of dilations.