Answer:
• 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:
[tex]\begin{gathered} m\widehat{QR}=2\times m\angle P \\ 180\degree=2\times m\angle P \\ \text{ Divide both sides by 2} \\ \frac{2\times m\angle P}{2}=\frac{180}{2} \\ m\angle P=90\degree \end{gathered}[/tex]
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.
[tex]UV=\sqrt{8^2+6^2}=\sqrt{100}=10[/tex]
Thus, divide the diameter by 2.
(I)The radius of the circle, r= 5 units.
(ii)Area
[tex]\begin{gathered} Area=\pi r^2 \\ \approx3.14\times5^2 \\ \operatorname{\approx}78.5\text{ square units} \end{gathered}[/tex]
The area of the circle is 78.5 square units.
