Let's analyze the given right triangle so that we can find the values of x and y.
Consider the bigger triangle. We know the hypotenuse, an expression for the altitude, an expression for the length of the shorter leg, and the length of a segment of the hypotenuse. Therefore, we can use the corollaries of the Right Triangle Similarity Theorem to write two proportions. Let's do it!
Finding x
To find the value of x, we will use the following corollary of the Right Triangle Similarity Theorem.
Let's compare the theorem's corollary to the expressions in our figure. In our case, DB is 40, AD is 50-40= 10, and x is the altitude of the bigger triangle CD.
AD/CD=CD/DB
⇔
10/x=x/40
Now we can use the Cross Product Property to find the value of x.
Note that, when solving the equation, we only considered the principal root. The reason for this is that, since x represents a side length, it must be positive. Therefore, the value of x is 20.
Finding y
To find the value of y, we will use the following corollary.
In a similar manner, let's compare the second theorem's corollary to the expressions in our figure. Consider the bigger triangle. The length of the shorter leg is y and the hypotenuse is 50. Consider now the smaller triangle at the right of the given diagram. Its hypotenuse is y, and the length of its shorter leg is 50-40= 10.
AB/AC=AC/AD
⇔
50/y=y/10
Now we can use the Cross Product Property to find the value of y.
Again, when solving the equation, we only considered the principal root. The reason for this is that, since y also represents a side length, it must be positive. The value of y is 10sqrt(5).
