line segment bd is a diameter of circle e.
what is the measure of arc bc?

- 39^
- 78^
- 102^
- 129^

Question

line segment bd is a diameter of circle e.
what is the measure of arc bc?

- 39^
- 78^
- 102^
- 129^

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [ "Inscribed Angle Theorem", "Thales's Theorem" ], "current_concepts": [ "Inscribed Angle Theorem", "Thales's Theorem", "Circle Geometry" ] } ## Identify the properties of the inscribed triangle $$ \begin{aligned} &\text{Since } BD \text{ is a diameter of circle } E, \text{ the inscribed angle } \angle BCD \text{ subtends a semicircle.}\ &\angle BCD = 90^\circ \end{aligned} $$ ## Find the measure of the third angle in triangle BCD $$ \begin{aligned} &\angle DBC = 51^\circ\ &\angle BDC = 180^\circ - \angle BCD - \angle DBC\ &\angle BDC = 180^\circ - 90^\circ - 51^\circ = 39^\circ \end{aligned} $$ ## Calculate the measure of arc BC $$ \begin{aligned} &\text{By the Inscribed Angle Theorem, the measure of the intercepted arc is twice the inscribed angle.}\ &\text{m}\widehat{BC} = 2 \times \angle BDC\ &\text{m}\widehat{BC} = 2 \times 39^\circ = 78^\circ \end{aligned} $$

O $39^\circ$

O $78^\circ$

O $102^\circ$

O $129^\circ$

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

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QUESTION IMAGE

Question

Explanation:

Identify the properties of the inscribed triangle

Find the measure of the third angle in triangle BCD

Calculate the measure of arc BC

Answer: