jm and kn are the diameters of $\odot p$. find the measures of each arc. then, identify each arc as a major arc, minor arc, or semicircle.

$m \widehat{mn} = $
$m \widehat{lm} = $
$m \widehat{kln} = $
$m \widehat{jln} = $
$m \widehat{jn} = $
$m \widehat{njl} = $

Question

jm and kn are the diameters of $\odot p$. find the measures of each arc. then, identify each arc as a major arc, minor arc, or semicircle.

$m \widehat{mn} = $
$m \widehat{lm} = $
$m \widehat{kln} = $
$m \widehat{jln} = $
$m \widehat{jn} = $
$m \widehat{njl} = $

Accepted Answer

{ "quality": "clear", "question_count": 6, "skills_matched": [ "step_cot", "table_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [ "Arc Measure", "Central Angles", "Arc Classification" ], "current_concepts": [ "Arc Measure", "Central Angles", "Arc Classification" ] } ## Find central angles and basic arc measures $$ \begin{aligned} &JM \text{ is a diameter } \implies m\angle JPM = 180^\circ\ &KN \text{ is a diameter } \implies m\angle KPN = 180^\circ\ &m\angle KPL = 74^\circ, \quad m\angle LPN = 180^\circ - 74^\circ = 106^\circ\ &m\angle LPM = m\angle LPN - m\angle MPN = 106^\circ - 62^\circ = 44^\circ\ &m\angle JPK = m\angle JPM - m\angle KPL - m\angle LPM = 180^\circ - 74^\circ - 44^\circ = 62^\circ\ &m\angle JPN = m\angle KPN - m\angle JPK = 180^\circ - 62^\circ = 118^\circ \end{aligned} $$ ## Calculate requested arc measures $$ \begin{aligned} &m\widehat{MN} = m\angle MPN = 62^\circ\ &m\widehat{LM} = m\angle LPM = 44^\circ\ &m\widehat{KLN} = m\angle KPN = 180^\circ\ &m\widehat{JLN} = 360^\circ - m\widehat{JN} = 360^\circ - 118^\circ = 242^\circ\ &m\widehat{JN} = m\angle JPN = 118^\circ\ &m\widehat{NJL} = m\widehat{NJ} + m\widehat{JK} + m\widehat{KL} = 118^\circ + 62^\circ + 74^\circ = 254^\circ \end{aligned} $$ ## Classify each arc by measure $$ \begin{aligned} &\text{Measure } < 180^\circ \implies \text{minor arc}\ &\text{Measure } = 180^\circ \implies \text{semicircle}\ &\text{Measure } > 180^\circ \implies \text{major arc} \end{aligned} $$ | No. | Problem | Answer | |---|---|---| | 1 | $m\widehat{MN}$ | $62^\circ$, minor arc | | 2 | $m\widehat{LM}$ | $44^\circ$, minor arc | | 3 | $m\widehat{KLN}$ | $180^\circ$, semicircle | | 4 | $m\widehat{JLN}$ | $242^\circ$, major arc | | 5 | $m\widehat{JN}$ | $118^\circ$, minor arc | | 6 | $m\widehat{NJL}$ | $254^\circ$, major arc | { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Geometry", "Arc Measure" ] }

No.	Problem	Answer
2	\(m\widehat{LM}\)	\(44^\circ\), minor arc
3	\(m\widehat{KLN}\)	\(180^\circ\), semicircle
4	\(m\widehat{JLN}\)	\(242^\circ\), major arc
5	\(m\widehat{JN}\)	\(118^\circ\), minor arc
6	\(m\widehat{NJL}\)	\(254^\circ\), major arc

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QUESTION IMAGE

Question

Explanation:

Find central angles and basic arc measures

Calculate requested arc measures

Classify each arc by measure

Answer: