circle t has diameters $\overline{rp}$ and $\overline{qs}$. the measure of $\angle rtq$ is $12^\circ$ less than the measure of $\angle rts$.

what is the measure of $\widehat{qp}$?

$78^\circ$
$84^\circ$
$88^\circ$
$96^\circ$

Question

circle t has diameters $\overline{rp}$ and $\overline{qs}$. the measure of $\angle rtq$ is $12^\circ$ less than the measure of $\angle rts$.

what is the measure of $\widehat{qp}$?

$78^\circ$
$84^\circ$
$88^\circ$
$96^\circ$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Linear Pair Postulate", "Supplementary Angles", "Central Angle Theorem" ], "new_concepts": [], "current_concepts": [ "Linear Pair Postulate", "Supplementary Angles", "Central Angle Theorem", "Vertical Angles" ] } ## Set up the system of equations for the linear pair $$ \begin{aligned} &m\angle RTQ + m\angle RTS = 180^\circ \ &m\angle RTQ = m\angle RTS - 12^\circ \end{aligned} $$ ## Solve for the central angles $$ \begin{aligned} &(m\angle RTS - 12^\circ) + m\angle RTS = 180^\circ \ &2m\angle RTS = 192^\circ \ &m\angle RTS = 96^\circ \ &m\angle RTQ = 96^\circ - 12^\circ = 84^\circ \end{aligned} $$ ## Find the measure of arc QP $$ \begin{aligned} &m\angle QTP = m\angle RTS = 96^\circ \quad (\text{vertical angles}) \ &m\widehat{QP} = m\angle QTP = 96^\circ \end{aligned} $$

(A) 78°

(B) 84°

(C) 88°

(D) 96°

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

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QUESTION IMAGE

Question

Explanation:

Set up the system of equations for the linear pair

Solve for the central angles

Find the measure of arc QP

Answer: