Algebra Logical Pythagorean theorem help A wire is attached to the top of a pole. The pole is 2 feet shorter than the wire, and the distance from the wire on the ground to the bottom of the pole is 9 feet less than the length of the wire. Find the length of the wire and the height of the pole.
Hint: Use the pythagorean theorem then set up a quadratic equation equal to zero and solve. 
I got c=17 and c=5 but I don't know what the length of the wire is or the height of the pole.
 A: I think that the $c$ you solved for is the length of the wire, which is the hypotenuse of the triangle. You can use the information in the problem to find the length of the leg which lies on the ground, and then the Pythagorean Theorem tells you the height of the pole.
A: Your work is correct.  To complete the problem, you must consider what each answer implies about the lengths of the legs of the triangle.
Since the pole is $2$ feet shorter than the wire, it has length $c - 2~\text{ft.}$  Since the distance from the wire on the ground to the bottom of the pole is $9$ feet less than the length of the wire, the base of the right triangle has length $c - 9~\text{ft.}$  Hence, by the Pythagorean Theorem,
\begin{align*}
(c - 2~\text{ft.})^2 + (c - 9~\text{ft.})^2 & = c^2\\
c^2 - 4c~\text{ft.} + 4~\text{ft.}^2 + c^2 - 18c~\text{ft.} + 81~\text{ft.}^2 & = c^2\\
c^2 - 22c~\text{ft.} + 85~\text{ft.}^2 & = 0\\
(c - 5~\text{ft.})(c - 17~\text{ft.}) & = 0
\end{align*}
Hence, $c = 5~\text{ft.}$ and $c = 17~\text{ft.}$ satisfy the equation.  However, if $c = 5~\text{ft.}$, then the distance from the point where the wire touches the ground to the base of the pole is $c - 9~\text{ft.} = -4~\text{ft.}$, which is impossible since a distance cannot be negative.  Hence, $c = 17~\text{ft.}$, so the height of the pole is $c - 2~\text{ft.} = 15~\text{ft.}$
