Let's compare the theorems to the expressions in our figure. We will use the first expression which is the Geometric Mean Altitude Theorem. In our case, 4 is the length of a partial segment of the hypotenuse, 9 is the length of the other segment of the hypotenuse, and z is the length of the height of the triangle.
d/h=h/e
⇔
4/z=z/9
Now we can use the Cross Product Property to find the value of z.
Since a negative side length does not make sense, we only need to consider positive solutions. Therefore, we found that z=6.
Finding y
We've found that z=6. Let's add this information to our diagram.
Now, let's use the second expression from our figure which is the Geometric Mean Leg Theorem. We know that, 4+9= 13 is the length of the hypotenuse, 9 is the length of a partial segment of the hypotenuse, and y is the length of the leg that is adjacent to the partial segment.
c/a=a/e
⇔
13/y=y/9
Once again, we can use the Cross Product Property to find the value of y.
Since a negative side length does not make sense, we only need to consider positive solutions. Therefore, we found that y=3sqrt(13)≈ 10.8.
Finding x
Let's add obtained information to the diagram.
Finally, we can use the third expression from the figure which is the Geometric Mean Leg Theorem. We already know that 4 is the length of one segment of the hypotenuse, 13 is the length of the hypotenuse, and x is the length of the shorter leg of the triangle.
c/b=b/d
⇔
13/x=x/4
Once again, we will use the Cross Product Property to find the value of x.
