Question

1. \\(m\widehat{can} = 226^{\circ}\\). find \\(m\widehat{cn}\\).
2. \\(df = 30\\). find the circumference of \\(\odot e\\).
3. \\(m\widehat{dow} = 294^{\circ}\\). find \\(m\widehat{dw}\\).
4. \\(no = 24\\). find the circumference of \\(\odot n\\).
5. \\(uq = 13\\). find the circumference of \\(\odot u\\).
6. \\(on = 43\\). find \\(ot\\).
7. \\(is = 17\\). find the circumference of \\(\odot r\\).
8. \\(tv = 32\\). find \\(uv\\).

Explanation:

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<reasoning>

Find the measure of arc MCN

The measure of a full circle is \(360^\circ\). The major arc \(\widehat{CAN}\) has a measure of \(226^\circ\). The remaining arc \(\widehat{MCN}\) completes the circle:
\[
m\widehat{MCN} = 360^\circ - m\widehat{CAN} = 360^\circ - 226^\circ = 134^\circ
\]

Find the circumference of circle E

The diameter of circle \(E\) is \(DF = 30\). The formula for the circumference \(C\) using diameter \(d\) is:
\[
C = \pi d = 30\pi \approx 94.25
\]

Find the measure of arc MDW

The major arc \(\widehat{DOW}\) has a measure of \(294^\circ\). The remaining arc \(\widehat{MDW}\) completes the circle:
\[
m\widehat{MDW} = 360^\circ - m\widehat{DOW} = 360^\circ - 294^\circ = 66^\circ
\]

Find the circumference of circle N

The radius of circle \(N\) is \(NO = 24\). The formula for the circumference \(C\) using radius \(r\) is:
\[
C = 2\pi r = 2\pi(24) = 48\pi \approx 150.80
\]

Find the circumference of circle U

The radius of circle \(U\) is \(UQ = 13\). The formula for the circumference \(C\) is:
\[
C = 2\pi r = 2\pi(13) = 26\pi \approx 81.68
\]

Find the length of OT

The segment \(ON\) is a radius of circle \(O\), so \(ON = 43\). Since \(OT\) is also a radius of the same circle \(O\), its length must be equal to \(ON\):
\[
OT = ON = 43
\]

Find the circumference of circle R

The diameter of circle \(R\) is \(IS = 17\). The formula for the circumference \(C\) is:
\[
C = \pi d = 17\pi \approx 53.41
\]

Find the length of UV

The segment \(TV\) is a diameter of circle \(U\) with length \(TV = 32\). The segment \(UV\) is a radius of circle \(U\). The radius is half of the diameter:
\[
UV = \frac{TV}{2} = \frac{32}{2} = 16
\]
</reasoning>

<answer>

No.	Problem	Answer
10	\(DF=30\). Find the circumference of \(\odot E\).	\(30\pi \approx 94.25\)
11	\(m\widehat{DOW}=294^\circ\). Find \(m\widehat{MDW}\).	\(66^\circ\)
12	\(NO=24\). Find the circumference of \(\odot N\).	\(48\pi \approx 150.80\)
13	\(UQ=13\). Find the circumference of \(\odot U\).	\(26\pi \approx 81.68\)
14	\(ON=43\). Find \(OT\).	\(43\)
15	\(IS=17\). Find the circumference of \(\odot R\).	\(17\pi \approx 53.41\)
16	\(TV=32\). Find \(UV\).	\(16\)

</answer>

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Answer:

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<reasoning>

Find the measure of arc MCN

The measure of a full circle is \(360^\circ\). The major arc \(\widehat{CAN}\) has a measure of \(226^\circ\). The remaining arc \(\widehat{MCN}\) completes the circle:
\[
m\widehat{MCN} = 360^\circ - m\widehat{CAN} = 360^\circ - 226^\circ = 134^\circ
\]

Find the circumference of circle E

The diameter of circle \(E\) is \(DF = 30\). The formula for the circumference \(C\) using diameter \(d\) is:
\[
C = \pi d = 30\pi \approx 94.25
\]

Find the measure of arc MDW

The major arc \(\widehat{DOW}\) has a measure of \(294^\circ\). The remaining arc \(\widehat{MDW}\) completes the circle:
\[
m\widehat{MDW} = 360^\circ - m\widehat{DOW} = 360^\circ - 294^\circ = 66^\circ
\]

Find the circumference of circle N

The radius of circle \(N\) is \(NO = 24\). The formula for the circumference \(C\) using radius \(r\) is:
\[
C = 2\pi r = 2\pi(24) = 48\pi \approx 150.80
\]

Find the circumference of circle U

The radius of circle \(U\) is \(UQ = 13\). The formula for the circumference \(C\) is:
\[
C = 2\pi r = 2\pi(13) = 26\pi \approx 81.68
\]

Find the length of OT

The segment \(ON\) is a radius of circle \(O\), so \(ON = 43\). Since \(OT\) is also a radius of the same circle \(O\), its length must be equal to \(ON\):
\[
OT = ON = 43
\]

Find the circumference of circle R

The diameter of circle \(R\) is \(IS = 17\). The formula for the circumference \(C\) is:
\[
C = \pi d = 17\pi \approx 53.41
\]

Find the length of UV

The segment \(TV\) is a diameter of circle \(U\) with length \(TV = 32\). The segment \(UV\) is a radius of circle \(U\). The radius is half of the diameter:
\[
UV = \frac{TV}{2} = \frac{32}{2} = 16
\]
</reasoning>

<answer>

No.	Problem	Answer
10	\(DF=30\). Find the circumference of \(\odot E\).	\(30\pi \approx 94.25\)
11	\(m\widehat{DOW}=294^\circ\). Find \(m\widehat{MDW}\).	\(66^\circ\)
12	\(NO=24\). Find the circumference of \(\odot N\).	\(48\pi \approx 150.80\)
13	\(UQ=13\). Find the circumference of \(\odot U\).	\(26\pi \approx 81.68\)
14	\(ON=43\). Find \(OT\).	\(43\)
15	\(IS=17\). Find the circumference of \(\odot R\).	\(17\pi \approx 53.41\)
16	\(TV=32\). Find \(UV\).	\(16\)

</answer>

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