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 $DB$?

Once you know both $AD$ and $DB$, you know the entire hypotenuse $AB$. Use the fact that each leg of the original triangle is the geometric mean of the full hypotenuse and the hypotenuse segment next to that leg to write an equation for $BC^2$.

After you find $BC^2$, take the square root and simplify the radical by factoring out the largest perfect square factor.

In Desmos, type 144/8 to compute $DB$ from the equation $144 = 8x$. Note the value that Desmos gives; that is $DB$.

In a new Desmos expression, enter 8 + (previous_result) to represent $AB = AD + DB$. The output is the length of $AB$.

Now type sqrt( (previous_DB) * (previous_AB) ), replacing previous_DB and previous_AB with the numerical results you found. The value Desmos outputs is the length of $BC$; compare this number with the answer choices to see which one matches.

• $\angle C$ is the right angle, so $AB$ is the hypotenuse.
• The altitude from $C$ meets $AB$ at $D$, so $CD \perp AB$ and $AD$ and $DB$ are the two segments of the hypotenuse.
• You are given $CD = 12$ and $AD = 8$. Let $DB = x$ (an unknown for now). Then the full hypotenuse is $AB = AD + DB = 8 + x$.

We want to find the leg $BC$ of the original right triangle $ABC$.},{