Question

1. the circumference of \\(\odot i\\) is \\(21\pi\\). find \\(nx\\).
2. the circumference of \\(\odot a\\) is \\(36\pi\\). find \\(va\\).
3. \\(m\widehat{an} = 32^{\circ}\\) and \\(m\widehat{nf} = 94^{\circ}\\). find \\(m\widehat{af}\\).
4. \\(m\widehat{op} = 37^{\circ}\\). find \\(m\widehat{ps}\\).
5. the circumference of \\(\odot i\\) is \\(22\pi\\). find \\(ih\\).
6. \\(rn = 18\\). find \\(ra\\).
7. the circumference of \\(\odot i\\) is \\(16\pi\\). find \\(ht\\).
8. \\(m\widehat{fkn} = 168^{\circ}\\), \\(m\widehat{uf} = 34^{\circ}\\). find \\(m\widehat{un}\\).

Answer:

<pre_analysis>
{
"quality": "clear",
"question_count": 8,
"skills_matched": [
"step_cot",
"table_formatter",
"explain_then_answer",
"tutor",
"mastery_adaptive_brevity"
],
"needs_drawing": false,
"concepts_used": [
"Circle Circumference",
"Circle Radius and Diameter",
"Arc Measure"
],
"new_concepts": [
"Circle Chord Properties"
],
"current_concepts": [
"Circle Circumference",
"Circle Radius and Diameter",
"Arc Measure",
"Circle Chord Properties"
]
}
</pre_analysis>

<reasoning>
The image is oriented upside down. Let's first read and analyze each question by rotating the content mentally by 180 degrees.

Looking at the rotated questions:

Question 17

• Text: "17. The circumference of \(\odot I\) is \(21\pi\). Find \(NX\)."
• Diagram: A circle with center \(I\) and a segment \(NX\) passing through the center \(I\). Thus, \(NX\) is a diameter.
• Using the Circle Circumference and Circle Radius and Diameter knowledge points:

Using the Circle Circumference and Circle Radius and Diameter knowledge points
\[
C = \pi d \implies 21\pi = \pi \cdot NX \implies NX = 21
\]

Question 18

• Text: "18. The circumference of \(\odot A\) is \(36\pi\). Find \(VA\)."
• Diagram: A circle with center \(A\) and a segment \(VA\) from the center \(A\) to the boundary. Thus, \(VA\) is a radius.
• Using the Circle Circumference and Circle Radius and Diameter knowledge points:

Using the Circle Circumference and Circle Radius and Diameter knowledge points
\[
C = 2\pi r \implies 36\pi = 2\pi \cdot VA \implies VA = 18
\]

Question 19

• Text: "19. \(m\widehat{AN} = 32^\circ\) and \(m\widehat{NF} = 94^\circ\). Find \(m\widehat{AF}\)."
• Diagram: A circle with points \(A\), \(N\), and \(F\) on the boundary. The arc \(AF\) is composed of adjacent arcs \(AN\) and \(NF\).
• Using the Arc Measure knowledge point:

Using the Arc Measure knowledge point
\[
m\widehat{AF} = m\widehat{AN} + m\widehat{NF} = 32^\circ + 94^\circ = 126^\circ
\]

Question 20

• Text: "20. \(m\widehat{OP} = 37^\circ\). Find \(m\widehat{PS}\)."
• Diagram: A circle with center \(T\) and diameter \(OS\). Point \(P\) lies on the circle. Since \(OS\) is a diameter, the arc \(OPS\) is a semicircle, which measures \(180^\circ\).
• Using the Arc Measure knowledge point:

Using the Arc Measure knowledge point
\[
m\widehat{PS} = 180^\circ - m\widehat{OP} = 180^\circ - 37^\circ = 143^\circ
\]

Question 21

• Text: "21. The circumference of \(\odot I\) is \(22\pi\). Find \(IH\)."
• Diagram: A circle with center \(I\) and radius \(IH\).
• Using the Circle Circumference and Circle Radius and Diameter knowledge points:

Using the Circle Circumference and Circle Radius and Diameter knowledge points
\[
C = 2\pi r \implies 22\pi = 2\pi \cdot IH \implies IH = 11
\]

Question 22

• Text: "22. \(RN = 18\). Find \(RA\)."
• Diagram: A circle with center \(A\) and diameter \(RN\). Thus, \(RA\) is a radius.
• Using the Circle Radius and Diameter knowledge point:

Using the Circle Radius and Diameter knowledge point
\[
RA = \frac{RN}{2} = \frac{18}{2} = 9
\]

Question 23

• Text: "23. The circumference of \(\odot I\) is \(16\pi\). Find \(HT\)."
• Diagram: A circle with center \(I\) and diameter \(HT\).
• Using the Circle Circumference and Circle Radius and Diameter knowledge points:

Using the Circle Circumference and Circle Radius and Diameter knowledge points
\[
C = \pi d \implies 16\pi = \pi \cdot HT \implies HT = 16
\]

Question 24

• Text: "24. \(m\widehat{FKN} = 168^\circ\), \(m\widehat{UF} = 34^\circ\). Find \(m\widehat{UN}\)."
• Diagram: A circle with points \(F\), \(K\), \(N\), and \(U\) on the boundary. The entire circle is \(360^\circ\). The circle is divided into major arc \(\widehat{FKN}\) and arc \(\widehat{FNU}\) (which consists of \(\widehat{UN}\) and \(\widehat{UF}\)).
• Using the Arc Measure knowledge point:

Using the Arc Measure knowledge point
\[
m\widehat{FKN} + m\widehat{UF} + m\widehat{UN} = 360^\circ \implies 168^\circ + 34^\circ + m\widehat{UN} = 360^\circ \implies m\widehat{UN} = 158^\circ
\]
</reasoning>

<answer>

No.	Problem	Answer
18	The circumference of \(\odot A\) is \(36\pi\). Find \(VA\).	\(18\)
19	\(m\widehat{AN} = 32^\circ\) and \(m\widehat{NF} = 94^\circ\). Find \(m\widehat{AF}\).	\(126^\circ\)
20	\(m\widehat{OP} = 37^\circ\). Find \(m\widehat{PS}\).	\(143^\circ\)
21	The circumference of \(\odot I\) is \(22\pi\). Find \(IH\).	\(11\)
22	\(RN = 18\). Find \(RA\).	\(9\)
23	The circumference of \(\odot I\) is \(16\pi\). Find \(HT\).	\(16\)
24	\(m\widehat{FKN} = 168^\circ\), \(m\widehat{UF} = 34^\circ\). Find \(m\widehat{UN}\).	\(158^\circ\)

</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multi-part",
"knowledge_point": [
"Mathematics",
"Geometry",
"Circle Circumference"
]
}
</post_analysis>