5.1 Proportions
We have talked about figures, particularly triangles, being congruent. This means that they have the exact same size and shape. However, there are figures that have the same shape although not the same size. Two equilateral triangles, two circles or two squares will always have the same shape as each other, but one may be bigger than the other. When two figures have the same shape but not necessarily the same size, we say that they are similar. If two shapes are similar, then typically one of them is simply a larger version of the other. All corresponding angles are congruent, but all corresponding sides are in the same ratio. When we talk about a ratio between two numbers, we express that ratio as the quotient of the two numbers, that is, as a fraction. Another convenient way to express a ratio is to use a colon between the numbers. A proportion is an equality between two ratios. For example, if we say that the ratio of $a$ to $b$ is 3 to 4, we would write this proportion as $\frac{a}{b}=\frac{3}{4}$ or as $a:b=3:4$. The "colon" notation is particularly useful when using a ratio with more than two numbers. If the sides of a triangle are 5, 7, and 9, we could say that they are in the ratio $5:7:9$. (you saw this notation in some of the problems back in Exercises 3.5.)
There is algebra involved in dealing with proportions and we will now do a quick review of how to work with ratios and proportions. Remember that if you have two fractions that are equal to each other, then you can use the process of cross-multiplication to help you "solve" the equation. If $\frac{a}{b}=\frac{3}{4}$, then this statement is equivalent to $4a=3b$. This is one of the most useful properties of proportions that we will use.
There is also some special vocabulary that is often used when discussing two equal fractions. This is derived from the colon notation, where there are numbers written on the far outside, and numbers written in between. If $a:b=c:d$, then we call $a$ and $d$ the extremes of the proportion, while $b$ and $c$ are the means. We use the same language even when the proportion is written using fractions, so with the proportion $\frac{a}{b}=\frac{3}{4}$, we have the extremes $a$ and $4$, and the means $b$ and 3. Turning this proportion into the product statement $4a=3b$ is an example of the idea that the product of the means equals the product of the extremes. If the means in a proportion are equal — say $\frac{3}{x}=\frac{x}{12}$ — then we call $x$ the geometric mean or the mean proportional of 3 and 12. Typically in such a case, $x$ refers to the length of a segment, so the result is positive.
Example: Two numbers have a sum of 14 and are also in a ratio of 3 to 4. Find the numbers.
Solution: Since the sum of the numbers is 14, if one of the numbers is $x$ then the other number must be $14-x$. We can now write and solve the following proportion.
\begin{align*}
\frac{x}{14-x}&=\frac{3}{4}&&\text{By cross-multiplying we get}\\
4x&=3(14-x)&&\\
4x&=42-3x&&\\
7x&=42&&\\
x&=6&&
\end{align*}
Since one of the numbers is 6, the second number, $14-x$, is 8.
Two proportions are said to be equivalent if one can be turned into the other by appropriate algebraic manipulations. Some of these are very familiar. For example, $$\text{If } \frac{a}{b}=\frac{c}{d}\text{, then }\frac{b}{a}=\frac{d}{c}.$$ That is, the fractions can be turned up-side down and the equality still holds. Not quite as familiar is the following: $$\text{If }\frac{a}{b}=\frac{c}{d}\text{, then }\frac{a}{c}=\frac{b}{d}.$$ That is, the ratio of the numerators equals the ratio of the denominators. There are several less familiar ways that a proportion can be massaged. The first of these properties is that the proportion $$\frac{a}{b}=\frac{c}{d}\text{ is equivalent to, or can be written as, }\frac{a+b}{b}=\frac{c+d}{d}.$$ That is, you can add the denominators to the numerators and you'll have two new fractions that will be equal to each other. We can show this by adding 1 to both sides of the original equation and getting common denominators.
\begin{align*}
\frac{a}{b}&=\frac{c}{d}\\
\frac{a}{b}+1&=\frac{c}{d}+1\\
\frac{a}{b}+\frac{b}{b}&=\frac{c}{d}+\frac{d}{d}\\
\frac{a+b}{b}&=\frac{c+d}{d}
\end{align*}
A second property of proportions is that $$\text{If }\frac{a}{b}=\frac{c}{d}\text{, then }\frac{a}{b}=\frac{a+c}{b+d}.$$ The proof of this property will be left as an exercise.