Question 644·Medium·Math

The figure shows $A$, $O$, and $C$ on one straight line, so $AC$ is a diameter: it splits the circle into equal halves. Find the length of the upper semicircle by dividing the circumference by $2$: $\frac{160\pi}{2}=80\pi$.

An arc is a curved piece of the circle’s edge. The upper semicircle contains both marked arcs: $AB$ is labeled $3x$, and $BC$ is labeled $x$. Add their lengths: $3x+x=4x$.

Divide arc $AB$’s length by the upper semicircle’s length to get its fraction of that half: $\frac{3x}{4x}=\frac34$. The whole in this fraction is the upper semicircle. Apply an arc’s fraction to the same whole its labels cover.

Arc $AB$ is $\frac34$ of $80\pi$. Keep $\pi$ outside the arithmetic: type $\frac34\cdot80$ in Desmos. It shows $60$, the coefficient, or number multiplying $\pi$. So the minor arc length is $60\pi$. Choice B.