Radius and chord from an arc

Text

In a circle a $90^\circ$ arc $AB$ has length $12\pi$ cm. Find the radius of the circle and the chord $\overline{AB}$ subtending the arc.

Solution

From the arc length, with $\theta=90^\circ=\dfrac{\pi}{2}$: $L=r\theta\ \Longrightarrow\ 12\pi=r\cdot\frac{\pi}{2}\ \Longrightarrow\ r=24\ \text{cm}.$ The chord subtending a $90^\circ$ angle is the hypotenuse of a right isosceles triangle with legs $r$: $\overline{AB}=r\sqrt2=24\sqrt2\approx33.94\ \text{cm}.$ $\boxed{r=24\ \text{cm};\ \ \overline{AB}=24\sqrt2\approx33.94\ \text{cm}}$