ShmoopTube

00:05

Geometric Mean, a la Shmoop.

00:13

A triangle at the bar, named Aybeesee, or ABC for short, wants you to guess her height.

00:20

She's a supermodel... being a triangle, she's in really great... shape...

00:24

...so she's really proud of her height.

00:27

Her height is the length from point A, where her 90° angle is, to her hypotenuse at point D.

00:35

This line, called the altitude, splits her into two smaller right triangles.

00:40

Knowing that, there's a possibility that Aybeesee is really two baby triangles in a cocktail dress.

00:47

She gives you a hint....the length from C to D is 9 feet, and the length from D to B is 4.

00:55

What is her height?

00:57

If we want to find Aybeesee's height, we can use something called the geometric mean.

01:06

It's not as cruel as it sounds.

01:08

The geometric mean of two numbers is the square root of the product.

01:13

Sounds simple, but how do we use it?

01:16

First, let's take what we know.

01:18

We can use the similarity of the two triangles and make a proportion out of the sides.

01:26

The proportions tell us that CD over AD equals AD over BD.

01:33

Plugging in these lengths, we get that 9 over x equals x over 4.

01:38

Cross multiplying gives us: x squared equals 9 times 4

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Take the square root of both sides, and we get x equals plus or minus the square root of 9 times 4.

01:52

And look the square root of two numbers multiplied together...is the geometric mean.

02:00

Since Aybeesee can't have a negative height,

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we take the positive answer only... ...so her height is 6 feet.

02:09

Who knows?

02:10

Maybe she'll swoon at your knowledge of geometric means and pay for your Coca-Cola.

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