We are asked to find the value of x and y from the given illustration. Let's find them one at a time.
Finding x
Let's analyze the given right triangle so that we can find the value of x.
Consider the bigger triangle. The hypotenuse is 3+ 9, or 12, and the length of the larger leg is x. Consider now the smaller triangle at the right of the diagram. Its hypotenuse is x and the length of its larger leg is 9. Let's substitute these values in the second proportion that we wrote at the beginning of this solution.
AB/CB=CB/DB
⇒
12/x=x/9
Now we can use the Cross Product Property to find the value of x.
When solving the above equation, we only kept the principal root because x represents a side length. Therefore, it must be positive. We conclude that x is equal to 6sqrt(3).
Finding y
Let's look at the illustration and include the information we have just found.
Consider the smaller triangle at the left of the diagram. The lengths of the larger and shorter legs are y and 3, respectively. Consider now the smaller triangle at the right of the diagram. The lengths of its larger and shorter legs are 9 and y, respectively. We can substitute these values in the first proportion that we wrote.
CD/DB=AD/CD
⇒
y/9=3/y
Now we can use the Cross Product Property to find the value of y.
