c How does the angle measure relate to the arc measure?
A
a 40^(∘)
B
b mAD=194^(∘)
m∠ C=97^(∘)
C
c mAB=125^(∘)
AB≈ 17.5 inches
A≈ 69.8 in^2
Practice makes perfect
a The sum of the central angles in a circle equals 360^(∘). Therefore, by dividing 360^(∘) by 9 we can calculate the central angle.
360^(∘)/9=40^(∘)
b The arc mAD refers to the arc that does not pass through B and C. Let's highlight this arc in red and add the measure of ∠ B to the diagram.
Examining the diagram, we notice that ∠ B and ∠ C are both inscribed angles to the sameintercepted arc. This must mean that they have the same measure.
m∠ C= m∠ B ⇔ m∠ C = 97^(∘)
Also, the intercepted arc is always twice the measure of a corresponding inscribed angle. This must mean that mAD can be obtained by multiplying the measure of m∠ B by 2.
mAD =2(97^(∘))=194^(∘)
c Let's add the given information to the diagram. Also, AB refers to the minor arc highlighted below.
The intercepted arc has the same measure as its corresponding central angle. Therefore, if m∠ ACB=125^(∘) then mAB=125^(∘). To find the arc length, AB, we have to multiply the circumference of the circle with the ratio of the central angle to 360^(∘).
AB=2π(8)* 125^(∘)/360^(∘)≈ 17.5 inches
The arc length is about 17.5 inches. To find the area of the sector, we multiply the area of the circle with the ratio of the central angle to 360^(∘).
A=π(8)^2* 125^(∘)/360^(∘)≈ 69.8 in.^2
The area of the sector is about 69.8 inches^2.
