point p is the incenter of $\delta rst$

which must be true?

$\bigcirc\ \overline{cp} \cong \overline{rp}$
$\bigcirc\ \overline{bp} \cong \overline{pc}$
$\bigcirc\ \angle psc \cong \angle ptc$
$\bigcirc\ \angle bps \cong \angle apt$

Question

point p is the incenter of $\delta rst$

which must be true?

$\bigcirc\ \overline{cp} \cong \overline{rp}$
$\bigcirc\ \overline{bp} \cong \overline{pc}$
$\bigcirc\ \angle psc \cong \angle ptc$
$\bigcirc\ \angle bps \cong \angle apt$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Incenter Theorem" ], "new_concepts": [], "current_concepts": [ "Incenter Theorem", "Triangle Centers" ] } ## Identify the properties of the incenter $$ \begin{aligned} &\text{Incenter } P \text{ is the intersection of the angle bisectors of } \Delta RST.\ &\text{The distance from } P \text{ to each side of the triangle is equal.}\ &PB \perp RS, \quad PA \perp RT, \quad PC \perp ST\ &PA = PB = PC \end{aligned} $$ ## Evaluate the given options $$ \begin{aligned} &\overline{CP} \cong \overline{RP} \quad (\text{False, } P \text{ is not equidistant from vertices})\ &\overline{BP} \cong \overline{PC} \quad (\text{True, since } PB = PC)\ &\angle PSC \cong \angle PTC \quad (\text{False, angles are bisected but not necessarily equal to each other})\ &\angle BPS \cong \angle APT \quad (\text{False, these angles are not congruent in general}) \end{aligned} $$ ## Determine the correct statement $$ \overline{BP} \cong \overline{PC} $$

(A) $\overline{CP} \cong \overline{RP}$

(B) $\overline{BP} \cong \overline{PC}$

(C) $\angle PSC \cong \angle PTC$

(D) $\angle BPS \cong \angle APT$

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

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QUESTION IMAGE

Question

Explanation:

Identify the properties of the incenter

Evaluate the given options

Determine the correct statement

Answer: