Exercise 58 Page 648

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

We know the length of both segments 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 y

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, 12 is the length of one segment of the hypotenuse, 4 is the length of the other segment of the hypotenuse, and y is the length of the height of the triangle. d/h=h/y ⇔ 12/y=y/4 Now 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=4sqrt(3)≈ 6.9.

Finding z

We have found that y=4sqrt(3). Let's add this information to our diagram.

Now, let's use the Geometric Mean Leg Theorem. We already know that 4 is the length of one segment of the hypotenuse, 4+12= 16 is the length of the hypotenuse, and z is the length of one of the legs of the triangle. c/a=a/e ⇔ 16/z=z/4 Once again, we can use the Cross Product Property to find the value of z.

Again, since a negative side length does not make sense, we only need to consider positive solutions. Therefore, we found that z=8.

Finding x

Let's add obtained information to the diagram.

Finally, we can use the Geometric Mean Leg Theorem once more. We already know that 12 is the length of one segment of the hypotenuse, 16 is the length of the hypotenuse, and x is the lenght of one of the legs. c/b=b/d ⇔ 16/x=x/12 Let's find the value of x!