Geometric Mean Examples

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

The definition of the geometric mean is the positive square root of the product of two numbers. Here, we're square-rooting the product of 7 and 7.

Multiplying 7 × 7 gives us 49.

Taking the positive square root of 49 gives us 7. That means the geometric mean of 7 and 7 is (you guessed it) 7.

m = 7

Find the geometric mean of 9 and 25 using proportions.

Setting up the proportion, this is what we get, where the geometric mean is x. As you can see, it's on the bottom of the first fraction and the top of the second.

Cross-multiplying gives us numbers on one side and the square of the geometric mean on the other.

x² = 9 × 25

Multiplying 9 × 25 gives us 225 and taking the positive square root of 225 gives us 15.

x = 15

Find the length of the altitude of ∆XYZ.

Our altitude is a. If we were on a plane, it would be 38,000 feet but thank goodness we aren't. First, we have to figure out how to set up our proportion. Looking at the two smaller triangles, we'll put the long sides on the top and the short sides on the bottom.

Now, we use the geometric mean. If we cross-multiply, we get this equation.

a² = 3.2 × 1.8

Plugging 3.2 × 1.8 into our calculator gives us 5.76. Hopefully our answer won't be a big ugly decimal. If we take the square root, we get a nice, clean number: 2.4.