## Right Triangles, Altitudes,
and the Geometric Mean

Click to check Practice Problems from
Right Triangles and Similarity

### Review

Recall the geometric mean between two positive numbers a,b
is the number x such that

Now remember the right triangle with an altitude drawn from
the hypotenuse that we previously worked with:

First consider the two triangles formed by the altitude:

We showed earlier that these are similar triangles. Therefore
we have the following ratio:

**Activity:** On a sheet of
paper, **solve for AD**. In terms of this solution what can
you say about AD in terms of CD and DB (**Hint**: the word geometric mean should be used)?

Looking at our original right triangle ABC, we see AD is the
altitude the altitude drawn from the vertex of the right angle
to the hypotenuse:

**Right Triangle Altitude Theorem Part
a: **The measure of the **altitude **drawn f**rom
the vertex of the right angle** of a right triangle to its hypotenuse
is the **geometric mean between the measures of the two segments
of the hypotenuse**.

In terms of our triangle, this theorem simply states what we
have already shown:

since AD is the altitude drawn from the right angle of our
right triangle to its hypotenuse, and CD and DB are the two segments
of the hypotenuse.

**Activity: **Open the GSP
Sketch by clicking on GSP Sketch below. Find the area of the triangle
(use the geometric mean).

**GSP Sketch**

Now let's look at our right triangle again:

In your notebook, list the **three similar triangles,**
and next to each triangle, list its **hypotenuse.**

**Activity:** Using the rules
of similar triangles click on the **GSP
Sketch **and fill in the proportions. Print the sketch and
add it to your notebook. You can also save it to your computer.

Considering our triangle again:

From the previous activity we found the following:

and

On your paper use words (including the geometric mean) to describe
the two relations above. Hint: you may want to use cross multiplication.

**Right Triangle Altitude Theorem Part
b: **If the altitude is drawn to the hypotenuse of a
right triangle, each leg of the right triangle is the geometric
mean of the hypotenuse and the segment of the hypotenuse adjacent
to the leg.

Click the lightbulb to practice what you have learned.