Quadrilateral OPQR is inscribed in circle N, as shown below. Which of the following could be used to calculate the measure of ∠P
QR? Circle N is shown with a quadrilateral OPQR inscribed inside it. Angle O is labeled x plus 16. Angle P is not labeled. Angle Q is labeled 6x minus 4. Angle R is labeled 2x plus 16.
1 answer:
You might be interested in
(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:
and since <TRO=pi/2, it must be that
(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:
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:
and so the angle <OST is
Towards determining |TS|:
Use cosine:
Finally, we can determine the arc length QT: