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At a Glance - Geometric Mean

To find altitudes of unruly triangles, we can just use the geometric mean, which actually isn't mean at all. It's quite nice. Just multiply two numbers together and take the square root. The positive number is the answer. That's it.

So if you're ever at a bar (drinking a Coca-Cola or chocolate milk, of course) and a right triangle asks you to find the geometric mean of 4 and 16, you won't break a sweat. It'll look like this:

Bam. We're done. Now, the reason this is helpful is because it lets the triangle know you care, and it comes from a proportion that looks like this:

If we cross-multiply, we can solve for the geometric mean and it'll give us the definition.

m2 = 4 × 16

Take the square root, and you're back to square one. Easy peasy.

Why would a right triangle ask you about geometric means? More importantly, how would a right triangle ask you about geometric means?

Say that a triangle at the bar, named Aybeesee (or ABC for short), wants you to guess her height. She's a supermodel, so she's really proud of her height.

Well, her height is the length from point A, where her 90° angle is, down toward her hypotenuse at point D. This line, called the altitude, splits her into two smaller right triangles. Ouch.

Knowing that, there's a possibility that Aybeesee is really two baby triangles in a cocktail dress. Well, maybe not.

They might be different sizes, but these three right triangles are all similar. The "identical angles" kind of similar, not the "Mary-Kate and Ashley Olsen" kind of similar.

If we want to find Aybeesee's height, we can use the similarity of the triangles and make a proportion out of the sides. Aybeesee's height is the short side of one baby triangle and the long side of the other baby triangle.


Compare the short side to the long side of each triangle and set them equal to each other. It'll look like this:

Whaddayaknow? It's the same proportion as the geometric mean. So you can find out Aybeesee's height no problem. She'll swoon at your knowledge of geometric means and maybe even pay for your Coca-Cola. Maybe.

Two things to remember about these proportions:

  1. The type of side is separated by the fraction.
  2. Different triangles are separated by the equal sign.

That means if a hypotenuse is in the numerator of one fraction, another hypotenuse should be on the numerator of the other. If the shortest side of the first triangle is in the denominator, ditto for the second. Basically, this:

The next step is actually solving these bad boys. We're still doing math, so why don't we throw some numbers up in here?

Sample Problem

What is the length of the altitude of this triangle?

If we substitute those numbers into our proportion, we can solve for the length of the altitude. Hopefully it's a smallish number, or else we may get lightheaded.

Cross-multiplying gives us:

x2 = 9 × 4

We've found the length of the altitude. Not exactly the altitude of Mt. Everest, but it's a start. We can use these proportions to solve for more than just the length of the altitude. If we have the right information, we can find the length of any side of any triangle. Whoa.

Sample Problem

Take this triangle, for example. We want to find the length of the biggest triangle's hypotenuse. We already know that 8 is the smaller piece (but we won't tell it because we don't want to hurt its feelings), so all we have to do is find out what the bigger piece is. Setting up our proportion is a good place to start.

The unknown piece, which we'll call a (because it refused to tell us its real name), is the long side of the top triangle, and the altitude is the short side. The altitude is also the long side of the bottom triangle, and the smaller piece of the hypotenuse is the short side. Our proportion should look like this:

12 × 12 = a × 8

144 = 8a

If we divide both sides by 8, we can figure out the mysterious a's real identity: 18. Unfortunately, that's not our answer. We wanted to know the length of the hypotenuse of the biggest triangle, but 18 is only a piece of it. Thankfully, we know the length of the other piece so all that's left is adding them together.

8 + a = 8 + 18 = 26

Done. Feel free to do a victory dance. Don't forget to stretch first, though.

Aside from winning over flirty right triangles at bars, what good are geometric means to us? If you go to museums (and we're sure a cultured student like yourself frequents them), you can use geometric means to look at Van Goghs and Monets.

When admiring these beautiful (and some not-so-beautiful) works of art, the best distance to stand is the distance of the altitude. That way, you'll be far enough away to see the whole painting without having to move your eyes around but close enough to see the details. Who says math isn't artistic?

Example 1

Find the geometric mean of 7 and 7 using the definition of the geometric mean.

Example 2

Find the geometric mean of 9 and 25 using proportions.

Example 3

Find the length of the altitude of ∆XYZ.

Exercise 1

What is the geometric mean of 1 and 4?

Exercise 2

What is the geometric mean of 15 and 3?

Exercise 3

What is the length of the altitude of this triangle?

Exercise 4

You just bought a 20-foot tall monster TV screen that'll probably burn your retinas before you can finish an episode of Adventure Time. In fact, you have to put it on the floor because otherwise it won't fit in your room. If your eyes are at 5.5 feet and your TV is on the floor, how far away should you stand from the TV so that you can see the whole screen at once? 

Exercise 5

Find the length of x.

Exercise 6

An animatronic bat is being built—because let's face it, who doesn't want an animatronic bat?—with wings in the shape of right triangles. The dimensions are 18 inches for the underside and 15 inches on top. To support its lifelike flight, a beam must be inserted into each wing at the altitude. How long must this beam be, to the nearest tenth of an inch?

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