in the diagram below, $\overline{ab}$ and $\overline{bc}$ are tangent to $\odot o$. what is the measure of $\widehat{ac}$?

a. $90^\circ$
b. $68^\circ$
c. $112^\circ$
d. $136^\circ$

Question

in the diagram below, $\overline{ab}$ and $\overline{bc}$ are tangent to $\odot o$. what is the measure of $\widehat{ac}$?

a. $90^\circ$
b. $68^\circ$
c. $112^\circ$
d. $136^\circ$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Angles Outside Circle Theorem", "Arc Measure" ], "new_concepts": [], "current_concepts": [ "Angles Outside Circle Theorem", "Arc Measure", "Tangent Line" ] } ## Identify the given values and target unknown Using the *Arc Measure* knowledge point * Angle outside the circle: $m\angle B = 68^\circ$ * Major arc measure: $m\widehat{ADC} = 248^\circ$ * Target unknown: Minor arc measure $m\widehat{AC}$ ## Apply the theorem for angles outside a circle Using the *Angles Outside Circle Theorem* knowledge point $$ m\angle B = \frac{1}{2}(m\widehat{ADC} - m\widehat{AC}) $$ ## Solve for the minor arc measure Using the *Angles Outside Circle Theorem* knowledge point $$ \begin{aligned} 68^\circ &= \frac{1}{2}(248^\circ - m\widehat{AC}) \ 136^\circ &= 248^\circ - m\widehat{AC} \ m\widehat{AC} &= 248^\circ - 136^\circ \ m\widehat{AC} &= 112^\circ \end{aligned} $$

A. $90^\circ$

B. $68^\circ$

C. $112^\circ$

D. $136^\circ$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "Angles Outside Circle Theorem" ] }

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QUESTION IMAGE

Question

Explanation:

Identify the given values and target unknown

Apply the theorem for angles outside a circle

Solve for the minor arc measure

Answer: