Let's analyze the given right triangle so that we may find the values of x, y, and z.
We know the length of both segments of the hypotenuse and expressions for other sides of a triangle. Therefore, we can use a corollary of the Right Triangle Altitude Theorem to write a proportion.
We will find x, y, and z one at a time.
Finding x
Let's compare the theorem's corollary to the expressions in our figure. In our case, 9 is the length of a partial segment of the hypotenuse, 16 is the length of the other segment of the hypotenuse, and x is the length of the height of the triangle.
x/h=h/y
⇔
9/x=x/16
Now we can solve the above equation using the Properties of Equality, to find the value of x.
Since a negative side length does not make sense, we only need to consider positive solutions. Therefore, we found that x=12.
Finding z
We have found that x=12. Let's add this information to our diagram.
Now, let's use the second part of our corollary. We know that 9+16=25 is the length of the hypotenuse, 16 is the length of a partial segment of the hypotenuse, and z is the length of the leg that is adjacent to the partial segment.
c/a=a/y
⇔
25/z=z/16
Once again, we can use the Properties of Equality 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=20.
Finding y
Let's add the obtained information to the diagram.
Finally, we can use the third part of the corollary. We already know that 9 is the length of one segment of the hypotenuse, 20 is the length of the longer leg, and 12 is the length of the height of the triangle.
b/a=x/h
⇔
y/20=9/12
Once again, we will use the Properties of Equality to find the value of y.
