Question

A. Assume that the diagram at right is not drawn to scale. If the measure of the arc QR = 180°, then what is m

Answer 1

• A. m∠P=90°

,

• B. r=5 units, Area=78.5 square units

Explanation:

Part A

• The measure of arc QR = 180°

,

• Angle P is the angle subtended by arc QR at the circumference of the circle.

From one of the circle's theorem:

Therefore:

`\begin{gathered} m\widehat{QR}=2* m\angle P \\ 180\degree=2* m\angle P \\ \text{ Divide both sides by 2} \\ (2* m\angle P)/(2)=(180)/(2) \\ m\angle P=90\degree \end{gathered}`

The measure of angle P is 90 degrees.

Part B

By definition, the measure of an arc is the measure of its central angle. If the measure of the arc is 180 degrees as in part (A) above, then arc QR is a semicircle.

Thus, a special relationship from part(a) is that the angle subtended by the diameter at the circumference is 90 degrees.

Using this concept:

• If UV is a diameter; and

,

• TU=6

,

• TV=8

The diagram is as follows:

We can find the length of the diameter UV using the Pythagorean Theorem.

`UV=√(8^2+6^2)=√(100)=10`

Thus, divide the diameter by 2.

(I)The radius of the circle, r= 5 units.

(ii)Area

The area of the circle is 78.5 square units.