neefd help fast
Hello, AHDDM!
We are given: $\displaystyle AD = 12$ and $\displaystyle DC - BD = 18\;\;\Rightarrow\;\; DC = BD + 18$
Let $\displaystyle BD = x$, then $\displaystyle DC = x + 18$
$\displaystyle \Delta ABC$ is inscribed in a semicircle.
. . Hence, $\displaystyle \Delta ABC$ is a right triangle.
$\displaystyle AD$ is the altitude to the hypotenuse of a right triangle.
Theorem: The altitude to the hypotenuse of a right triangle is
. . the mean proportional of the two segments of the hypotenuse.
Hence: .$\displaystyle x(x+18) \,=\,12^2$
We have the quadratic: .$\displaystyle x^2 + 18x - 144\:=\:0$
. . which factors: .$\displaystyle (x - 6)(x + 24) \:=\:0$
. . and has the positive root: .$\displaystyle x = 6$
Therefore: .$\displaystyle BD = 6,\;\;DC = 24\quad\Rightarrow\quad BC = 30$ cm.