Exercise 56 Page 648

Let's analyze the given right triangle so that we may find the values of the variables.

We know the length of the longer leg of the triangle, the length of a segment of the hypotenuse, and the expressions for other sides of triangle. Therefore, we can use the Geometric Mean Altitude Theorem and the Geometric Mean Leg Theorem to write a proportion.

We will find x, y, and z one at a time.

Finding z

Let's compare the theorems to the expressions in our figure. We can use the second expression which is the Geometric Mean Leg Theorem. In our case, z+10 is the length of the hypotenuse, 10 is the length of a partial segment of the hypotenuse, and 12 is the length of the leg that is adjacent to the partial segment. c/a=a/e ⇔ z+10/12=12/10 Now we can use the Cross Product Property to find the value of z.

Finding x

We have found that z=4.4. Let's add this information to our diagram.

Now, let's use the Geometric Mean Altitude Theorem, which is the first expression. We already know that 4.4 is the length of one segment of the hypotenuse, 10 is the length of the other segment of the hypotenuse, and x is the length of the height of the triangle. d/h=h/e ⇔ 4.4/x=x/10 Once again, we can use the Cross Product Property 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=2sqrt(11)≈ 6.6.

Finding y

Let's add obtained information to the diagram.

Finally, we can use the Geometric Mean Leg Theorem once more. We already know that 4.4+10= 14.4 is the length of the hypotenuse and that the length of the shorter segment of the hypotenuse is 4.4. c/b=b/d ⇔ 14.4/y=y/4.4 Once again, we will use the Cross Product Property to find the value of y.