Let's analyze the given right triangle so that we may find the value of x.
We know the length of the hypotenuse, an expression for the length of a segment of the hypotenuse, and the expression for the length of the leg adjacent to this segment. Therefore, we can use a corollary of the Right Triangle Similarity Theorem to write a proportion.
Corollary 2 to Theorem 7-3
The altitude to the hypotenuse of a right triangle separates the hypotenuse so that the length of each leg of the triangle is the geometric mean of the length of the hypotenuse and the lengths of the segment of the hypotenuse adjacent to the leg.
We can also visualize this corollary.
Let's compare the theorem's corollary to the expressions in our figure. In our case, 12 is the length of the hypotenuse, x is the length of a partial segment of the hypotenuse, and x+3 is the length of the leg that is adjacent to the partial segment.
AB/AC=AC/AD
⇔
12/x+3=x+3/x
Now we can use the Cross Product Property to find the value of x.
