Right triangle $ABC$ has leg lengths $AB=9$ and $BC=12$. Including $\overline{AB}$ and $\overline{BC}$, how many line segments with integer length can be drawn from vertex $B$ to a point on hypotenuse $\overline{AC}$?
$AC=\sqrt{9^2+12^2}=15$ by the Pythagorean theorem.
The altitude from $B$ is $BD=\dfrac{9\cdot12}{15}=7.2$ by theorem 11.
As a point moves from $D$ to $A$, its distance from $B$ increases continuously from $7.2$ to $9$, so the integer lengths $8,9$ each occur once; from $D$ to $C$ it increases from $7.2$ to $12$, giving $8,9,10,11,12$.
