8.3. Using Similar Right Triangles http://www.ck12.org
A practical application of the geometric mean is to find the altitude of a right triangle.
Example 7:Find the value ofx.
Solution:Using similar triangles, we have the proportion
shortest leg of smallest 4
shortest leg of middle 4=
longer leg of smallest 4
longer leg of middle 4
9
x=
x
27
x^2 = 243
x=√
243 = 9
√
3
In Example 7,^9 x= 27 x is in the definition of the geometric mean. So, the altitude is the geometric mean of the two
segments that it divides the hypotenuse into.
Theorem 8-6:In a right triangle, the altitude drawn from the right angle to the hypotenuse divides the hypotenuse
into two segments. The length of the altitude is the geometric mean of these two segments.
Theorem 8-7:In a right triangle, the altitude drawn from the right angle to the hypotenuse divides the hypotenuse
into two segments. The length of each leg of the right triangle is the geometric mean of the lengths of the hypotenuse
and the segment of the hypotenuse that is adjacent to the leg.
Inotherwords
Theorem 8-6:BCAC=DCAC
Theorem 8-7:BCAB=ABDBandDCAD=ADDB
Both of these theorems are proved using similar triangles.
Example 8:Find the value ofxandy.