No, if the diagonals were congruent then the “kite” would be a square. Since the two pairs of congruent sides cannot be congruent to each other (i.e., they must be distinct), the diagonals will have different lengths.
Chapter 7: Similarity
Ratios and Proportions
Learning Objectives
Write and simplify ratios.
Formulate proportions.
Use ratios and proportions in problem solving.
Introduction
Words can have different meanings, or even shades of meanings. Often the exact meaning depends on the context in which a word is used. In this chapter you’ll use the word similar.
What does similar mean in ordinary language? Is a rose similar to a tulip? They’re certainly both flowers. Is an elephant similar to a donkey? They’re both mammals (and symbols of national political parties in the United States!). Maybe you’d rather say that a sofa is similar to a chair? In loose terms, by similar we usually mean that things are like each other in some way or ways, but maybe not the same.
Similar has a very precise meaning in geometry, as we’ll see in upcoming lessons. To understand similar we first need to review some basic skills in ratios and proportions.
Using Ratios
A ratio is a type of fraction. Usually a ratio is a fraction that compares two parts. “The ratio of to ” can be written in several ways.
to
Example 1
Look at the data below, giving sales at Bagel Bonanza one day.
Bagel Bonanza Monday Sales
Type of bagel Number sold
Plain
Cinnamon Raisin
Sesame
Garlic
Whole grain
Everything
a) What is the ratio of the number of cinnamon raisin bagels sold to the number of plain bagels sold?
Ratio of cinnamon raisin to plain , or to .
Note: Depending on the problem, ratios are often written in simplest form. In this case the ratio can be reduced or simplified because .
b) What is the ratio, in simplest form, of the number of whole grain bagels sold to the number of "everything" bagels sold?
Ratio of whole grain to everything , or to .
c) What is the ratio, in simplest form, of everything bagels sold to the number of whole grain bagels sold?
Answer: This ratio is just the reciprocal of the ratio in b. If the ratio of whole grain to everything is, , or to , then the ratio of everything to whole grain is, , or to .
d. What is the ratio, in simplest form, of the number of sesame bagels sold to the total number of all bagels sold?
First find the total number of bagels sold: .
Ratio of sesame to total sold , or to .
Note that this also means that , or , of all the bagels sold were sesame.
In some situations you need to write a ratio of more than two numbers. For example, the ratio, in simplest form, of the number of cinnamon raisin bagels to the number of sesame bagels to the number of garlic bagels is (or before simplifying).
Example 2
A talent show features only dancers and singers.
The ratio of dancers to singers is .
There are performers in all.
How many singers are there?
There is a whole number so that the total number of each group can be represented as
Since there are dancers and singers in all,
The number of dancers is . The number of singers is . It’s easy to check these answers. The numbers of dancers and singers have to add up to , and they have to be in a ratio.
Check: . The ratio of dancers to singers is , or to .
Proportions
A proportion is an equation. The two sides of the equation are ratios that are equal to each other. Proportions are often found in situations involving direct variation. A scale drawing would make a good example.
Example 3
Leo uses a scale drawing of his barn. He recorded actual measurements and the lengths on the scale drawing that represent those actual measurements.
Barn dimensions Actual length Length on scale drawing
Door opening
Interior wall
Water line
?
a) Since he is using a scale drawing, the ratio of actual length to length on the scale drawing should be the same all the time. We can write two ratios that should be equal. This is the proportion below.
Is the proportion true?
We could write the fractions with a common denominator. One common denominator is .
The proportion is true.
b) Depending on how you think, you might have written a different proportion. You could say that the ratio of the actual lengths must be the same as the ratio of the lengths on the scale drawing.
This proportion is also true. One nice thing about working with proportions is that there are several proportions that correctly represent the same data.
c) What length should Leo use on the scale drawing for the water line?
Let represent the scale length. Write a proportion.
If two fractions are equal, and they have the same denominator, then the numerators must be equal.
The scale length for the water line is .
Note that the scale for this drawing can be expressed as to , or to .
Proportions and Cross Products
Look at example 3b above.
is true if and only if .
In the proportion, and are called the means (they’re in the middle); and are called the extremes (they’re on the ends). You can see that for the proportion to be true, the product of the means must equal the product of the extremes . Both products equal .
It is easy to generalize this means-and-extremes rule for any true proportion.
Means and Extremes Theorem or The Cross Multiplication Theorem
Cross Multiplication Theorem: Let and be real numbers, with and
If then .
The proof of the cross multiplication theorem is example 4. The proof of the converse is in the Lesson Exercises.
Example 4
Prove The Cross Multiplication Theorem: For real numbers , and with and If , then .
We will start by summarizing the given information and what we want to prove. Then we will use a two-column proof.
Given: and are real numbers, with and and
Prove:
Statement Reason
1. and are real numbers, with and
1. Given
2.
2. Given
3.
3. , identity property of multiplication
4.
4. Commutative property of multiplication
5. or
5. If equal fractions have the same denominator, then the numerators must be equal
This theorem allows you to use the method of cross multiplication with proportions.
Lesson Summary
Ratios are a useful way to compare things. Equal ratios are proportions. With the Means-and-Extremes Theorem we have a simple but powerful method for solving any proportion.
Points to Consider
Proportions are very “forgiving”—there are many different ways to write proportions that are equivalent to each other. There are hints of some of these in the Lesson Exercises. In the next lesson, we’ll prove that these proportions are equivalent.
You know about figures that are congruent. But many figures that are alike are not congruent. They may have the same shape, even though they are not the same size. These are similar figures; ratios and proportions are integral to defining and understanding similar figures.
Review Questions
The votes for president in a club election were:
Write each of the following ratios in simplest form. votes for Milhone to votes for Suarez
votes for Cho to votes for Milhone
votes for Suarez to votes for Milhone to votes for Cho
votes for Suarez or Cho to total vot
es
Use the diagram below for exercise 2.
Write each of the following ratios in simplest form.
The measures of the angles of a triangle are in the ratio . What are the measures of the angles?
The length and width of a rectangle are in a ratio. The area of the rectangle is . What are the length and width?
Prove the converse of Theorem 7-1: For real numbers and with, and .
Given: and are real numbers, with and and
Prove:
Which of the following statements are true for all real numbers and and ? If then .
If then .
If then .
If then .
Solve each proportion for .
Shawna drove and used of gas. At that rate, she would use of gas to drive . Write a proportion that could be used to find the value of .
Solve the proportion you wrote in exercise 8. How much gas would Shawna expect to use to drive
Rashid, Leon, and Maria are partners in a company. They divide the profits in a ratio, with Rashid getting the largest share and Leon getting the smallest share. In the company had a total profit of . How much profit did each person receive?
Review Answers
and
Statement Reason
A. and with and , and . A. Given
B. B. Multiplication Property of Equality
C. C. Arithmetic
D. D. Arithmetic
E. E. , identity property of equality
No
Yes
Yes
No
or
or equivalent
. At that rate she would use about of gas.
Rashid gets , Leon gets , and Maria gets .
Properties of Proportions
Learning Objectives
Prove theorems about proportions.
Recognize true proportions.
Use proportions theorems in problem solving.
Introduction
The Cross Multiplication Theorem is the basic, defining property of proportions. Whenever you are in doubt about whether a proportion is true, you can always check it by cross multiplication. Additionally, there are also a number of “sub-theorems” about proportions that are useful to apply for solving problems. In each case the sub-theorem is easy to prove using cross multiplication.
Properties of Proportions
Technically speaking, the theorems in this lesson are not called sub-theorems. The formal term is corollary. The word corollary is rather loosely defined in mathematics. Basically, a corollary is a theorem that follows quickly, easily, and directly from another theorem—in this case from the Cross multiplication Theorem.
The corollaries in this section are not absolutely essential—you could always go back to using cross multiplication. But there may be times when the corollaries make things quicker or easier, so it’s good to have them if and when they are needed.
Cross Multiplication Corollaries
Below are three corollaries that are immediate results of the Cross Multiplication Theorem and the fundamental laws of algebra.
Corollaries 1, 2, and 3 of The Cross Multiplication Theorem
If and and , then ....
.
.
.
In words.
A true proportion is also true if you “swap” the “means.”
A true proportion is also true if you “swap” the “extremes.”
A true proportion is also true if you “flip” it upside down.
Example 1
Look at the diagram below.
Suppose we’re given that .
We know , since
Here are some other proportions that must also be true by corollaries 1-3.
Two Additional Corollaries to the Cross Multiplication Theorem
Here we have two more corollaries to the Cross Multiplication Theorem. The “if” part of these theorems is the same as above. So the given in each proof remains the same too.
Corollary 4:
If , and , and , then
Proof.
Statement Reason
1. , and , and 1. Given
2. 2. Cross Multiplication Theorem
3. 3. Distributive Property
4. 4. Distributive Property
5. 5. Substitution
6. 6. Substitution
7. 7. Cross Multiplication Theorem
This second theorem is nearly the same as the previous,
Corollary 5:
If and and , then
The proof of this corollary is in the Lesson Exercises.
Example 2
Suppose we’re given that again, as in example 1.
Here are some other proportions that must also be true, and the theorems that guarantee them.
Lesson Summary
Proportions were probably not new to you in this lesson; you may have studied them in previous courses. What probably is new is the larger structure of theorems and corollaries that serve as tools for working with proportions.
The most basic fact about proportions is the Cross Multiplication Theorem:
assuming and . The corollaries in this lesson are really just variations on the Cross Multiplication Theorem. They may be useful in problems, but we could always revert back to Cross Multiplication if we had to.
Some people find proportions nice to work with, because there are so many different—and correct—ways to write a given proportion, as you saw in the corollaries. It sometimes seems that you would really have to work at it to write a proportion that is not equivalent to the proportion you are given!
Points to Consider
As we move ahead we will meet important concepts that require the use of ratios and proportions. Proportions are mandatory for understanding the geometric meaning of similar. Later when we work with transformations and scale factors, ratios will also be useful.
Finally, one proof of the Pythagorean Theorem relies on proportions.
Review Questions
Given that . For each of the following, write “true” if the proportion must be true. Otherwise write “false.”
Prove: If , then .
Prove Corollary 5 to the Cross Multiplication Theorem.
Review Answers
False
True
False
True
True
False
True
True
True
False
Statement Reason
A. A. Given
B. B. Cross Multiplication Theorem
C. C. Distributive Property
D. D. Substitution
E. E. Distributive Property
F. F. Cross Multiplication Theorem
Statement Reason
A. A. Given
B. B. Cross Multiplication Theorem
C. C. Distributive Property
D. D. Substitution
E. E. Distributive Property
F. F. Cross Multiplication Theorem
Similar Polygons
Learning Objectives
Recognize similar polygons.
Identify corresponding angles and sides of similar polygons from a statement of similarity.
Calculate and apply scale factors.
Introduction
Similar figures, rectangles, triangles, etc., have the same shape. Same shape, however, is not a precise enough term for geometry. In this lesson, we’ll learn a precise definition for similar, and apply it to measures of the sides and angles of similar polygons.
Similar Polygons
Look at the triangles below.
The triangles on the left are not similar because they are not the same shape.
The triangles in the middle are similar. They are all the same shape, no matter what their sizes.
The triangles on the right are similar. They are all the same shape, no matter how they are turned or what their sizes.
Look at the quadrilaterals below.
The quadrilate
rals in the upper left are not similar because they are not the same shape.
The quadrilaterals in the upper right are similar. They are all the same shape, no matter what their sizes.
The quadrilaterals in the lower left are similar. They are all the same shape, no matter how they are turned or what their sizes.
Now let’s get serious about what it means for figures to be similar. The rectangles below are all similar to each other.
These rectangles are similar, but it’s not just because they’re rectangles. Being rectangles guarantees that these figures all have congruent angles. But that’s not enough. You’ve seen lots of rectangles before, some are long and narrow, others are more blocky and closer to square in shape.
The rectangles above are all the same shape. To convince yourself of this you could measure the length and width of each rectangle. Each rectangle has a length that is exactly twice its width. So the ratio of length-to-width is for each rectangle. Now we can make a more formal statement of what similar means in geometry.
Two polygons are similar if and only if:
CK-12 Geometry Page 26