Exercise 32 Page 645

Let's divide the given triangle into two right triangles.

We will deal with these triangles one at a time.

Triangle 1

Notice that this is a right triangle with an acute angle that measures 60^(∘). Therefore, by the Triangle Angle Sum Theorem the measure of the third angle must be 30^(∘).

We have a 30^(∘)-60^(∘)-90^(∘) triangle. In this type of triangle, the length of the longer leg is sqrt(3) times the length of the shorter leg. This will let us find the value of y.

Triangle 2

Our second right triangle has an angle that measures 30^(∘). Therefore, by the Triangle Angle Sum Theorem the measure of the third angle is 60^(∘).

We have a 30^(∘)-60^(∘)-90^(∘) triangle. In this type of triangle, the length of the hypotenuse is twice the length of the shorter leg. x= 2 * 3sqrt(3) ⇔ x=6sqrt(3)