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Similar Triangles - The Basics of Similarity

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The Basics of Similarity

Oh, shapes, how do we compare thee? Let us count the ways.

1. You could be congruent.
2. You could be similar.
3. You could be totally unrelated.

Okay, that's enough.

Let's talk more about being similar. What does it mean for two shapes to be similar? Perhaps we can explain it with an analogy. We know what you're thinking: "Analogies? This is math, not English class." Tough noodles for you.

How about this? Congruence is to Popeye as similarity is to Wimpy. If congruence dined regularly on canned spinach, then similarity would gladly pay you Tuesday for a hamburger today.

In other words, similarity is a little weaker than congruence. We could say that congruent shapes have congruent angles and congruent sides. We could also say that pumpkin pie is a necessity on any Thanksgiving dinner table, but that's a given, so we won't waste our breath.

Similar shapes, on the other hand, have congruent angles, but proportional sides. That is, if we wrote ratios to compare the side lengths, the ratios would be equivalent.

  • Ratio

    Beep, beep, back the truck up. Let's quickly remember that a ratio is a comparison of two related quantities found by dividing one quantity by another. To write a ratio, you can either write the two numbers as a fraction, write the two numbers with a colon (no, not that kind of colon! Gross!) in the middle, or write the word "to" in between the two numbers.

    Let's say we have 3 blue stars, 5 red stars, and 2 gold stars. There are oodles of ratios we can write about the number of stars we have. We could write a ratio comparing blue stars to red stars (35), red stars to gold stars (52), or gold stars to all stars (210).

    It's important to know that the ratio of gold stars to red stars (25) is not the same as the ratio of red stars to gold stars (52). If we turn the words around, the numbers will follow.

    Sample Problem

    What is the ratio of cats to dogs?

    The ratio of cats to dogs will name how many cats there are compared to how many dogs there are. Since there are three cats and five dogs, our choices are ⅗, 3:5, and 3 to 5. Cover your eyes, cat lovers. The dogs win this time.

    Sample Problem

    Mrs. Greene's geometry class has 14 girls and 12 boys. What is the ratio of girls to total students?

    Since there are 14 girls and 26 total students, the ratio is 14:26. We could also write it as 14 to 26 or . Or, we could get really fancy and say 7:13, 7 to 13, or .

    Whaaaa?! Yes, it's true. In very much the same way that we can manipulate fractions, we can also manipulate ratios. We can reduce them to their simplest terms, convert them to percentages, and, most importantly for our purposes today, find equivalent ratios. When you do find two equivalent ratios, you've got yourself a proportion. But hold your horses, we'll get to that in a minute.

    Sample Problem

    At the local park, children have the choice of spinning until they puke on the merry-go-round or giving themselves a wicked case of vertigo on the swings. The little thrill-seekers choose the swings over the merry-go-round by a margin of 3 to 1. Last week, 100 kids played at the park. How many of them swung themselves silly?

    Saying, "a margin of 3 to 1," is like saying, "a ratio of 3 to 1." Even more helpful, it's like saying, "3 out of 4," or three-fourths, or ¾. So, out of 100 kids on the playground, 75 of them swung themselves silly.

  • Proportion

    proportion is a statement of the equality of two ratios. Unlike the amateur portion who tells us that he kept that status just so he could compete in the portion Olympics, a proportion tells us when two ratios are equivalent.

    Let's dive a little deeper, shall we? Here we have a small school of fish. There's one fish, there's two fish, there's a red fish, and hey, there's a blue fish. We're a regular Dr. Seuss.

    The ratio of red to blue is , while the ratio of orange to green is . Because  is equivalent to , these ratios are proportional.

    What about comparing red fish to green fish and blue fish to orange fish? Well, the ratio of red fish to green fish is  while the ratio of blue fish to orange fish is . These fractions are not equivalent, so the ratios are not proportional.

    Sample Problem

    Spongebob and Patrick posted new photos on their Facebook timelines. Spongebob got 7 likes and 4 comments on his photo, while Patrick got 14 likes and 9 comments on his photo. Are the ratios of likes to comments on each photo proportional?

    The ratio of likes to comments on Spongebob's post is 7 to 4, whereas Patrick's friends responded in a ratio of 14 to 9. The ratios 7 to 4 and 14 to 9 are not equivalent, which means they're not proportional.

    Because proportions are equivalent, we can use cross-multiplication to solve them for any missing parts. Cross-multiplication, you're probably reminding your friends right now, is when you multiply the numerator on the left side by the denominator on the right side, which equals the left denominator times the right numerator.

    If we had the ratios  and , we could cross-multiply to find that 3 × 10 = 5 × 6. This handy little trick will help us solve proportion problems where one of the four numbers is missing.

    Let's say that you're a zoologist working on a groundbreaking formula for something epic, like, a magic powder that will bring unicorns back from extinction. (Yeah, there's an awful lot wrong with that statement, but let's just go with it.)

    We know that it will take, among other things, both seaborgium and hafnium (they're real, look 'em up) in a ratio of 5 parts Sg to 8 parts Hf. We have an unlimited supply of hafnium, but only 15 grams of seaborgium. If we want to use up all of the available seaborgium (so that we can bring back as many unicorns as possible), how much hafnium will we need to use?

    To solve this problem, and save a few unicorns, we can set up a proportion.

    Use cross-multiplication to solve for x.

    5x = 8 × 15
    5x = 120
    x = 120 ÷ 5
    x = 24

    We'll need to use 24 grams of hafnium. Bring on the unicorns!

    Sample Problem

    Find the value of n that makes this statement a proportion.

    We'll solve for n using cross-multiplication. That is, 3 × 28 = 7n. Or, 84 = 7n. Divide both sides by 7 to find 12 = n. The proportion would be .

    Sample Problem

    The ratio of sapphires to rubies and the ratio of rubies to total gemstones are proportional. We know we have 1 sapphire and 3 rubies. If the rest are diamonds, how many diamonds do we have?

    Set up a proportion.

    Cross-multiply to find g × 1 = 3 × 3, or g = 9. There are 9 total gemstones. Since we already know that 4 of them aren't diamonds, we know that 5 of them are. Let's hope they're obscenely big diamonds. Hint, hint.

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