(a) <TOR=pi/3 radians
To determine <TOR we use the fact that in the right-angled triangle ORT we know two sides:
|OT|=radius=8cm and |OR|=radius/2=4cm
and can use the sine:
[tex]\sin \angle OTR=\frac{r/2}{r}=\frac{1}{2}\implies \angle OTR =\frac{\pi}{6}[/tex]
and since <TRO=pi/2, it must be that
[tex]\angle TOR =\pi-\frac{\pi}{2}-\frac{\pi}{6}=\frac{\pi}{3}[/tex]
(b) The arc length is approximately 7.255 cm
In order to calculate the arc length QT, we need to first determine the length |ST| and the angle <OST.
Towards determining angle <OST:
[tex]\angle SOT = \pi - \angle TOR = \pi - \frac{\pi}{3} = \frac{2}{3}\pi[/tex]
Next, draw a line connecting P and T. Realize that triangle PTS is right-angled with <PTS=pi/2. This follows from the Thales theorem. Since R is a midpoint between P and O, it follows that the triangles ORT and PRT are congruent. So the angles <PTR and <OTR are congruent. Knowing <PTS we can determine angle <OTS:
[tex]\angle OTR \cong \angle PTR=\frac{\pi}{6}\implies\angle OTS=\angle PTS -\angle PTR -\angle OTR\\\angle OTS = \frac{\pi}{2}-\frac{\pi}{6}-\frac{\pi}{6}=\frac{\pi}{6}[/tex]
and so the angle <OST is
[tex]\angle OST = \pi - \angle TOS - \angle OTS = \pi -\frac{2}{3}\pi - \frac{1}{6}\pi=\frac{\pi}{6}[/tex]
Towards determining |TS|:
Use cosine:
[tex]\cos \angle OST =\frac{|RS|}{|ST|}\implies |ST|=\frac{\frac{3}{2}r}{\cos \frac{\pi}{6}}=\frac{12\cdot 2}{\sqrt{3}}=8\sqrt{3}cm[/tex]
Finally, we can determine the arc length QT:
[tex]QT = {\angle OST}\cdot |ST|=\frac{\pi}{6}\cdot 8 \sqrt{3}=\frac{4\pi}{\sqrt{3}}\approx 7.255cm[/tex]
