question 19 of 40

a student inscribes a triangle within a semicircle. what is the measure of $\angle xyz$?

a. $60^\circ$
b. $45^\circ$
c. $90^\circ$
d. $30^\circ$

Question

question 19 of 40

a student inscribes a triangle within a semicircle. what is the measure of $\angle xyz$?

a. $60^\circ$
b. $45^\circ$
c. $90^\circ$
d. $30^\circ$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Semicircle", "Inscribed Angle Theorem" ], "new_concepts": [ "Thales's Theorem" ], "current_concepts": [ "Semicircle", "Inscribed Angle Theorem", "Thales's Theorem" ] } ## Identify the inscribed angle and intercepted arc $$ \begin{aligned} &\text{Arc } XZ = 180^\circ \quad (\text{since } XZ \text{ is the diameter of the semicircle})\ &\angle XYZ \text{ is an inscribed angle intercepting arc } XZ \end{aligned} $$ ## Apply the Inscribed Angle Theorem $$ \begin{aligned} &m\angle XYZ = \frac{1}{2} \cdot m(\text{Arc } XZ)\ &m\angle XYZ = \frac{1}{2} \cdot 180^\circ = 90^\circ \end{aligned} $$

A. $60^\circ$

B. $45^\circ$

C. $90^\circ$

D. $30^\circ$

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

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QUESTION IMAGE

Question

Explanation:

Identify the inscribed angle and intercepted arc

Apply the Inscribed Angle Theorem

Answer: