Notice that x and y are the legs of the given right triangle, and z is a partial segment of the hypotenuse divided by the altitude.
Finding z
Since we know the lengths of the altitude and of a partial segment of the hypotenuse, we will use the Geometric Mean (Altitude) Theorem to find the value of z.
We need to compare the theorem to the expressions in our figure. In our case, sqrt(2) is the length of the altitude, and z and 3sqrt(2) are the lengths of the partial segments of the hypotenuse.
AD/CD = CD/DB
⇔ z/sqrt(2) = sqrt(2)/3sqrt(2)
Now, we can find the value of z.
Using a calculator, we can express z as about 0.5.
Finding x and y
Let's go back to the given figure.
Since we know the lengths of both partial segments of the hypotenuse, we will use the Geometric Mean (Leg) Theorem to find the values of x and y.
We will start by finding the value of x, which corresponds to CB on this figure.
CB = sqrt(DB * AB)
⇕
x = sqrt(3sqrt(2)(3sqrt(2)+sqrt(2)/3))
To find the value of x, we will evaluate the right-hand side.
Using a calculator, we can rewrite x as about 4.5. Following the same reasoning, we can find y, which corresponds to AC.
AC = sqrt(AD * AB)
⇕
y=sqrt(sqrt(2)/3 (3sqrt(2)+sqrt(2)/3))
Finally, we can evaluate the right-hand side to find the value of y.
