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Similar Right Triangles side lengths

Video, Interactive Applet and Examples

Video Tutorial on Similar Right Triangle
Geometric Mean example The mean proportion is any value that can be expressed just the way that 'x' is in the proportion on the left.
equation 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)
equation 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)
picture of the geometric mean altitude
To better understand how the altitude of a right triangle acts as a mean proportion in similar triangles, look at the triangle below with sides a, b and c and altitude H.

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HTML5 Applet

See Applet on its own page.
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Below is a picture of the many similar triangles created when you drop the altitude from a right angle of a right triangle.
Diagram of Geomoetric mean
Find the corresponding sides:
Answer
AD &

AD & Find the sides that correspond with AC
Answer


AC &

AC &
Find the sides that correspond with BC
Answer
BC &

BC &

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 proportion 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 .)

general saas
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)



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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.

example 2 picture

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.



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