mathschallenge.net

Problem

A vertical pole $AB$ measuring $5$ metres snaps at point $C$. The pole remains in contact at $C$ and the top of the pole touches the ground at point $T$, a distance of $3$ metres from $A$.

Find the length $AC$, the point where the pole snapped.

---

Solution

Let $AC = x$, so $BC = CT = 5 - x$.

Using the Pythagorean Theorem,

$$\begin{eqnarray}(5 - x)^2 & = & x^2 + 3^2\\25 - 10x + x^2 & = & x^2 + 9\\\therefore 10x & = & 16\end{eqnarray}$$

Hence $AC = \frac{8}{5}$ metres.