A dilation is a transformation that has a center and a scale factor. The center is a point and the scale factor governs how much the figure stretches or shrinks.
Think about watching a round balloon being inflated, and focusing on the point exactly in the middle of the balloon. The balloon stretches outwards from this point uniformly. So for example, if a circle is drawn around the point, this circle will grow as the balloon stretches away from the points.
Dilation with center at point and scale factor ,
Given a point that is from point . The image of for this dilation is the point that is collinear with and and from , the center of dilation.
Example 1
The center of dilation is , and the scale factor is .
Point is from . To find the image of point , we go from along to locate , the image of . Point is three times as far from as is , and , , and are collinear.
Note: The scale factor is . The length from to is “stretched” three times as long as the length from to .
Example 2
The center of dilation is , and the scale factor is .
Point is from , as in example 1. To find the image of point , we go from along to locate , the image of . Point is as far () from as is (), and , , and are collinear.
Note: The scale factor is . The length from to “shrinks” to as long as the length from to .
Example 3
is a rectangle. What are the length, width, perimeter, and area of ?
The center of a dilation is , and the scale factor is . What are the length, width, perimeter, and area of
Point is the same as point . is , and is .
In
Note: The perimeter of is the perimeter of , but the area of is the area of .
As the following diagram shows, four rectangles congruent to fit exactly into .
Coordinate Notation for Dilations
We can work with dilations on a coordinate grid. To simplify our work, we’ll study dilations that have their center of dilation at the origin.
Triangle in the diagram below is dilated with scale factor .
Triangle is the image of .
Notice that each side of is as long as the corresponding side of . Notice also that , and the origin are collinear. Thus is also true of , and the origin, and of , and the origin.
This leads to the following generalization
Generalization: Points (the image of ), and the origin are collinear for any point in a dilation. You can prove the generalization in the Lesson Exercises.
How do we know that a dilation is a similarity transformation? We would have to establish that lengths of segments are proportional and that angles are congruent. Let’s attack these requirements through the distance formula and slopes.
Let , and be points in a coordinate grid. Let a dilation have center at the origin and scale factor .
Part 1: Proportional Side Lengths
Let’s look at the lengths of two segments, , and .
According to the distance formula,
and
What does this say about a segment and its image in a dilation? It says that the image of a segment is another segment the length of the original segment. If a polygon had several sides, each side of the image polygon would be the length of its corresponding side in the original polygon.
Conclusion: If a polygon is dilated, the corresponding sides of the image polygon and the original polygon are proportional. So half the battle is over.
Part 2: Congruent Angles
Let’s look at the slopes of the sides of two angles, and .
Since and have the same slope, they are parallel. The same is true for and . We know that if the sides of two angles are parallel, then the angles are congruent. This gives us:
Conclusion: If a polygon is dilated, the corresponding angles of the image polygon and the original polygon are congruent. So the battle is now over.
Final Conclusion: If a polygon is dilated, the original polygon and the image polygon are similar, because they have proportional side lengths and congruent angles. A dilation is a similarity transformation.
Lesson Summary
Dilations round out our study of geometric transformations. Unlike translations, rotations, and reflections, dilations are not congruence transformations. They are similarity transformations. If a dilation is applied to a polygon, the image is a similar polygon.
Points to Consider
We limited our study of dilations to those that have positive scale factors. To explore further, you might experiment with negative scale factors.
Tech Note - Geometry Software
Use your geometry software to explore dilations with negative scale factors.
Exploration 1
Plot two points.
Select one of the points as the center of dilation.
Use 2 for the scale factor.
Find the image of the other point.
Repeat, but use a different value for the scale factor.
What seems to be true about the two images?
Exploration 2
Draw a triangle.
Select a point as the center of dilation. Use one vertex of the triangle, or draw another point for the center.
Use 2 for the scale factor.
Find the image of the triangle.
Repeat, but use a different value for the scale factor.
What seems to be true about the two images?
You can experiment further with different figures, centers, and scale factors.
Can you reach any conclusions about images when the scale factor is negative?
You may have noticed that if point is dilated, the center is , and the scale factor is , then the image of is on the same side of as is. If the scale factor is then the image of is on the opposite side of . You may have also also noticed that a dilation with a negative scale factor is equivalent to a dilation with a positive scale factor followed by a “reflection in a point,” where the point is the center of dilation.
This lesson brings our study of similar figures almost to a close. We’ll revisit similar figures once more in Chapter 10, where we analyze the perimeter and area of similar polygons. Some writers have used similarity concepts to explain why living things are the “right size” and why, for example, there are no human giants!
Review Questions
Use the diagram below for exercises 1 - 10.
and
A dilation has the indicated center and scale factor. Complete the table.
Center Scale Factor Given Point Image of Given Point
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? Midpoint of
? Midpoint of
Copy the square shown below. Draw the image of the square for a dilation with center at the intersection of and scale factor .
A given dilation is a congruence transformation. What is the scale factor of the dilation?
Imagine a dilation with a scale factor of . Describe the image of a given point for this dilation.
Let and be two points in a coordinate grid, . Prove that and are collinear.
A dilation has its center at the origin and a scale factor of . Let be the point . If is the image of , and is the image of , what are the coordinates of ?
Review Answers
Center Scale Factor Given Point Image of Given Point Answer
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? or
? Midpoint of or
? Midpoint of
Small square centered in big square, each side of big square side of small square
The image of any point is the point that is the center of dilation.
Let be the origin .
Since the segments have common endpoints and the same slope, they are collinear.
Self-Similarity (Fractals)
Lear
ning Objectives
Appreciate the concept of self-similarity.
Extend the pattern in a self-similar figure.
Introduction
In this lesson you’ll learn about patterns called fractals that have self-similarity. Instead of using a formal definition, we’ll work with a few examples that give the idea of self-similarity. In each example you will be able to see that later stages in a pattern have a similarity relationship to the original figure.
Example 1
The Cantor Set
The pattern in the diagram below is called the Cantor Set, named for a creative mathematician of the late 1800s.
The pattern continues. Now let’s see why this pattern is called “self-similar.”
Look at the circled part of the pattern.
You can see that each part of Level 2 is similar to Level 1 with a scale factor of . The same relationship continues as each level is created from the level before it.
Example 2
Sierpinski Triangle
To construct a Sierpinski Triangle, begin with an equilateral triangle. (Actually, any triangle could be used.) This is the Start level.
Then connect the midpoints of the sides of the triangle. Shade in the central triangle.
This is Level 1.
Now repeat this process to create Level 2:
Connect the midpoints of the sides of each unshaded triangle to form smaller triangles.
Shade in each central triangle.
This is Level 2.
The pattern continues, as shown below:
To view some great examples of Sierpinski Triangles visit the following link:
Now let’s see how the Sierpinski Triangle is self-similar.
Look at the triangle that is outlined in the diagram above. Could you prove that the outlined pattern is similar to the pattern of Level 1? Because of this relationship, the Sierpinski Triangle is self-similar.
Tech Note - Geometry Software
Use geometry software to create the next level, or levels, of the Sierpinski Triangle.
Lesson Summary
Fractals and self-similarity are fairly recent developments in geometry. The patterns are interesting on their own, and they have been found to have applications in the study of many natural and human-made fields. Successive levels of a fractal pattern are all similar to the preceding levels.
Points to Consider
You may want to learn more about fractals. Use a search engine to find information about fractals on the Internet.
Review Questions
Use the Cantor Set to answer questions 1-6.
How many segments are there in Level 3?
If the segment in the Start level is long, how long is each segment in Level 2?
How many segments are there in Level 4?
How many segments are there in Level 10?
How many segments are there in Level ?
If the segment in the Start level is long, how long is each segment in Level ?
Use the Sierpinski Triangle to answer questions 7-13.
How many unshaded triangles are there in Level 2?
How many unshaded triangles are there in Level 3?
How many unshaded triangles are there in Level ?
Suppose the area of the Start level triangle is .
What is the total area of the unshaded part of Level 1?
What is the total area of the unshaded part of Level 2?
What is the total area of the unshaded part of Level ?
Explain how you know that the outlined part of Level 2 is similar to Level 1.
Tech Note - Geometry Software
Use geometry software to create the next level of the fractal pattern shown below.
all lines should be straight and all angles right
all lines should be straight and all angles right
Review Answers
The midsegments of a triangle divide it into four congruent triangles, each of which is similar to the original triangle.
All lines should be straight and all angles right.
Chapter 8: Right Triangle Trigonometry
The Pythagorean Theorem
Learning Objectives
Identify and employ the Pythagorean Theorem when working with right triangles.
Identify common Pythagorean triples.
Use the Pythagorean Theorem to find the area of isosceles triangles.
Use the Pythagorean Theorem to derive the distance formula on a coordinate grid.
Introduction
The triangle below is a right triangle.
The sides labeled and are called the legs of the triangle and they meet at the right angle. The third side, labeled is called the hypotenuse. The hypotenuse is opposite the right angle. The hypotenuse of a right triangle is also the longest side.
The Pythagorean Theorem states that the length of the hypotenuse squared will equal the sum of the squares of the lengths of the two legs. In the triangle above, the sum of the squares of the legs is and the square of the hypotenuse is .
The Pythagorean Theorem: Given a right triangle with legs whose lengths are and and a hypotenuse of length ,
Be careful when using this theorem—you must make sure that the legs are labeled and and the hypotenuse is labeled to use this equation. A more accurate way to write the Pythagorean Theorem is:
Example 1
Use the side lengths of the following triangle to test the Pythagorean Theorem.
The legs of the triangle above are and . The hypotenuse is . So, , and . We can substitute these values into the formula for the Pythagorean Theorem to verify that the relationship works:
Since both sides of the equation equal , the equation is true. Therefore, the Pythagorean Theorem worked on this right triangle.
Proof of the Pythagorean Theorem
There are many ways to prove the Pythagorean Theorem. One of the most straightforward ways is to use similar triangles. Start with a right triangle and construct an altitude from the right angle to the opposite sides. In the figure below, we can see the following relationships:
Proof.
Given: as shown in the figure below
Prove:
First we start with a triangle similarity statement:
Now, using similar triangles, we can set up the following proportion:
These are all true by the triangle similarity postulate.
and
Putting these equations together by using substitution,
factoring the right hand side,
but notice , so this becomes
We have finished proving the Pythagorean Theorem. There are hundreds of other ways to prove the Pythagorean Theorem and one of those alternative proofs is in the exercises for this section.
Making Use of the Pythagorean Theorem
As you know from algebra, if you have one unknown variable in an equation, you can solve to find its value. Therefore, if you know the lengths of two out of three sides in a right triangle, you can use the Pythagorean Theorem to find the length of the missing side, whether it is a leg or a hypotenuse. Be careful to use inverse operations properly and avoid careless mistakes.
Example 2
What is the length of in the triangle below?
Use the Pythagorean Theorem to find the length of the missing leg, . Set up the equation , letting and . Be sure to simplify the exponents and roots carefully, remember to use inverse operations to solve the equation, and always keep both sides of the equation balanced.
In algebra you learned that because, for example, . However, in this case (and in much of geometry), we are only interested in the positive solution to because geometric lengths are positive. So, in example 2, we can disregard the solution , and our final answer is .
Example 3
Find the length of the missing side in the triangle below.
Use the Pythagorean Theorem to set up an equation and solve for the missing side. Let and .
So, the length of the missing side is .
Using Pythagorean Triples
In example 1, the sides of the triangle were , and . This combination of numbers is referred to as a Pythagorean triple. A Pythagorean triple is three numbers that make the Pythagorean Theorem true and they are integers (whole numbers with no decimal or fraction part). Throughout this chapter, you will use other Pythagorean triples as well. For instance, the triangle in example 2 is proportional to the same ratio of . If you divide the lengths of the triangle in example 2 by two, you find the same proportion—. Whenever you find a Pythagorean triple, you can apply those ratios with greater factors as well. Finally, take note of the side lengths of the triangle in example 3—. This, too, is a Pythagorean triple. You can extrapolate that this ratio, multiplied by greater factors, will also yield numbers that satisfy the Pythagorean Theorem.
There are infinitely many Pythagorean triples, but a few of the most common ones and their multiples are:
Triple
Area of an Isosceles Triangle
There are many different applications of the Pythagorean Theorem. One way to use The Pythagorean Theorem is to identify the heights in isosceles triangles so you can calculate the area. The area of a triangle is half of the product of its base and its height (also called altitude). This formula is shown below.
CK-12 Geometry Page 29