Question 464·Easy·Math

No Desmos needed. A special-triangle ratio gives the exact lengths. The right angle and $30^\circ$ angle identify a 30-60-90 triangle. The sides across from $30^\circ$, $60^\circ$, and $90^\circ$ are the short leg, long leg, and hypotenuse, with lengths $x$, $x\sqrt3$, and $2x$.

Match each leg to the angle across from it. The $30^\circ$ angle at $A$ is across from $BC$, so $BC$, not $AC$, is the short leg. That makes $AC$ the long leg: $AC=x\sqrt3$.

$AB$ lies across from the right angle at $C$, so it's the hypotenuse. Its given length is $10$, which fills the $2x$ position in the ratio: $2x=10$.

Divide both sides by $2$ to find $x$: $x=5$.

Substitute $x=5$ into the long-leg length $AC=x\sqrt3$: $AC=5\sqrt3$. So the length of $AC$ is $5\sqrt3$. Choice C.