Notice that x is a partial segment of the hypotenuse divided by the altitude, and y and z are the legs of the given right triangle. We will find their values one at a time.
Finding x
Since we know the length of one partial segment of the hypotenuse, and the length of the altitude, we will use the Geometric Mean (Altitude) Theorem to find the value of x.
We want to compare the theorem to the expressions in our figure. In our case, 16 is the length of the altitude, and 8 and x are the lengths of the partial segments of the hypotenuse.
AD/CD = CD/DB
⇔ 8/16 = 16/x
Now, we can find the value of x.
Since we know the lengths of both partial segments of the hypotenuse divided by the altitude, we will use the Geometric Mean (Leg) Theorem to find the values of y and z.
We will start by finding the value of y, which corresponds to CB on the figure above.
CB = sqrt(DB * AB)
⇔ y=sqrt(32(32+8))
Now we can evaluate the right-hand side to find y.
Using a calculator, we can express y as about 35.8. Following the same reasoning, we can find z, which corresponds to AC.
AC = sqrt(AD * AB)
⇔ z=sqrt(8(32+8))
Finally, we can evaluate the right-hand side.
