Similar Right Triangles

Video Tutorial on Similar Right Triangle

	The mean proportionis any value that can be expressed just the way that 'x' is in the proportion on the left.
	In the proportion on the left 'x', is the geometric mean, we could solve for x by cross multiplying and going from there (more on that later)
	In the proportion on the left, '4', is the geometric mean

So what does this have to do with right similar triangles?

It turns out the when you drop an altitude (h in the picture below) from the the right angle of a right triangle, the length of the altitude becomes a geometric mean. This occurs because you end up with similar triangles which have proportional sides and the altitude is the long leg of 1 triangle (left side) and the short leg of the other similar triangle (right side in pic below)

To better understand how the altitude of a right triangle acts as a mean proportionin similar triangles, look at the triangle below with sides a, b and c and altitude H.

Below is a picture of the many similar triangles created when you drop the altitude from a right angle of a right triangle.

Find the corresponding sides:

Answer

AD & AB

AD & BD

---

Find the sides that correspond with AC

Answer

AC & AB

AC & BC

---

Find the sides that correspond with BC

Answer

BC & AB

BC & AC

Example Problem Types

Students usually have to solve 2 different core types of problems involving the geometric mean.

Problem Type #1) The altitude and hypotenuse

As you can see in the picture below, this problem type involves the altitude and 2 sides of the inner triangles ( these are just the two parts of the large outer triangle's hypotenuse) . This lets us set up a mean proportionequation involving the altitude and those two sides (see demonstration above if you need to be convinced that these are indeed corresponding sides of similar triangles .)

What is the length of the altitude below?

Give me a hint first

This problem is just example problem 1 above (solving for an altitude using the parts of the large hypotenuse)

Problem Type #2) hypotenuse, Leg and Side

Involves the hypotenuse of the large outer triangle, one its legs and a side from one of the inner triangles.

Find the value of x in the triangle below:

Give me a hint first

This problem is just example problem 2 because it involves the outer triangle's hypotenuse, leg and the side of an inner triangle.

Find the value of x in the triangle below:

Give me a hint first

This problem is just example problem 2 because it involves the outer triangle's hypotenuse, leg and the side of an inner triangle.

Find the value of x in the triangle below:

Give me a hint first

This problem is just example problem 1 because it involves the altitude and the two parts of the outer triangle's hypotenuse.