point $o$ is the center of the given circle. what is the measure of $\widehat{df}$?

a. $104^\circ$
b. $19^\circ$
c. $38^\circ$
d. $76^\circ$

Question

point $o$ is the center of the given circle. what is the measure of $\widehat{df}$?

a. $104^\circ$
b. $19^\circ$
c. $38^\circ$
d. $76^\circ$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Circle Theorems" ], "new_concepts": [ "Inscribed Angle Theorem" ], "current_concepts": [ "Circle Theorems", "Inscribed Angle Theorem" ] } ## Identify the given values and geometric relationships $$ \begin{aligned} &\text{Diameter: } DE \quad (\text{since line segment passes through center } O) \ &m\angle OEF = 38^\circ \ &OF = OE = \text{radius} \implies \triangle OEF \text{ is isosceles} \end{aligned} $$ ## Calculate the central angle $$ \begin{aligned} &m\angle OFE = m\angle OEF = 38^\circ \ &m\angle EOF = 180^\circ - (38^\circ + 38^\circ) = 104^\circ \end{aligned} $$ ## Find the measure of arc DF $$ \begin{aligned} &m\angle DOF + m\angle EOF = 180^\circ \ &m\angle DOF = 180^\circ - 104^\circ = 76^\circ \ &m\widehat{DF} = m\angle DOF = 76^\circ \end{aligned} $$

A. 104°

B. 19°

C. 38°

D. 76°

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "Inscribed Angle Theorem" ] }

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QUESTION IMAGE

Question

Explanation:

Identify the given values and geometric relationships

Calculate the central angle

Find the measure of arc DF

Answer: