Question 448·Hard·Math

Point $D$ divides the hypotenuse $AB$ into two pieces, so add them: $AB=AD+DB$. Substitute their lengths: $AB=8+18$. Add:

An altitude meets a side at a right angle. So triangles $ABC$ and $CBD$ each have a right angle, and they share angle $B$. They are similar, meaning they have the same shape. Match $AB$ with $BC$, and $BC$ with $DB$, to write the equal ratios:

Corresponding sides have equal ratios, so cross-multiply: $AB\cdot DB=BC^2$. A leg squared equals the whole hypotenuse times the piece that leg touches. Here $BC$ touches $DB$ at $B$; using $AD$ would find the other leg.

Use $AB=26$ and the piece next to $BC$, $DB=18$. Substitute them into the relationship:

Let $x$ stand for $BC$. Type $x_1^2\sim26\cdot18\{x_1>0\}$ in Desmos. The tilde asks Desmos to find $x_1$, and the restriction keeps the positive length. Under PARAMETERS, Desmos gives $x_1\approx21.633$. Use that decimal to check the size of your exact answer.

Since $BC$ is positive, take the positive square root: $BC=\sqrt{26\cdot18}$. Multiply inside: $BC=\sqrt{468}$. Since $468=36\cdot13$, pull out the perfect-square factor $36$: $BC=\sqrt{36\cdot13}$. Take the square root of $36$: $BC=6\sqrt{13}$. So $BC$ is $6\sqrt{13}$ units long. Choice B.