Answer to Question #126886 in Geometry for jungkookiee

Question

QR is a diameter of a circle,centre O,and P is a point on its circumference.
Given that PQR= 25°, calculate
a, POR
b, OPR
c, the value, correct to 2 significant figures of PQ/QR.

Answer 1

Consider the circle with center O and the diameter PQ

$$central \ angle=intercepted\ arc$$

The measure of each inscribed angle is exactly half the measure of its intercepted arc. 

In a circle, the measure of an inscribed angle is half the measure of the central angle with the same intercepted arc.

a. The angle $\angle PQR$ is an inscribed angle

$$m\angle PQR={1\over 2}m \overgroup{PR}$$

The angle $\angle POR$ is a central angle

$$m\angle POR=m \overgroup{PR}=2m \angle PQR=2\cdot25\degree=50\degree$$

b. We see that OP and OR are radii of the circle. Then $\Delta POR$ is the equilateral triangle

$$OP=OR, m\angle OPR=m\angle ORP$$

The three interior angles in a triangle will always add up to 180°

$$m\angle OPR+m\angle ORP+m\angle POR=180\degree$$

$$m\angle OPR+m\angle ORP=180\degree -m\angle POR$$

$$m\angle OPR=\dfrac{180\degree -m\angle POR}{2}$$

$$m\angle OPR=\dfrac{180\degree -50\degree}{2}=65\degree$$

c. An angle inscribed in a semicircle is a right angle. Then

$$m\angle QPR=90\degree$$

Right $\Delta QPR$

$$\cos\angle PQR=\dfrac{PQ}{QR}$$

$$\dfrac{PQ}{QR}=\cos25\degree\approx0.91$$