There is a special relationship in a right triangle: the altitude from the right angle to the hypotenuse is the geometric mean of the two segments of the hypotenuse. How can you write an equation using $CD$, $AD$, and $DB$?

After finding $DB$, add it to $AD$ to find $AB$. Then use $BC^2=DB\cdot AB$.

Take the square root of your value for $BC^2$ and factor the number inside the radical to identify a perfect-square factor.

Enter d=12^2/8 in Desmos. This calculates $DB$ using $CD^2=AD\cdot DB$.

Enter h=8+d to calculate $AB=AD+DB$.

Enter sqrt(d*h). This evaluates $\sqrt{DB\cdot AB}$, which is the length of $BC$. Compare the resulting value with the answer choices.

Since $CD$ is an altitude to the hypotenuse of a right triangle, its length is the geometric mean of the two hypotenuse segments:

Substitute the known values:

Thus,

The hypotenuse consists of segments $AD$ and $DB$, so

A leg of the original right triangle is the geometric mean of the full hypotenuse and the adjacent segment of the hypotenuse. Since $BC$ is adjacent to $DB$,

Therefore,