find the perimeter of the triangle.
$p =$

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{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [ "Tangent Segments Theorem", "Triangle Perimeter with Inscribed Circle" ], "current_concepts": [ "Tangent Segments Theorem", "Triangle Perimeter with Inscribed Circle" ] } ## Determine the lengths of the tangent segments from each vertex $$ \begin{aligned} &\text{Let the vertices of the triangle be } A \text{ (top), } B \text{ (bottom-left), and } C \text{ (bottom-right).}\ &\text{Let the points of tangency on sides } AB, BC, \text{ and } AC \text{ be } D, E, \text{ and } F \text{ respectively.}\ &CF = 5.6\ &AF = AC - CF = 14.6 - 5.6 = 9.0\ &AD = AF = 9.0\ &BD = AB - AD = 15.1 - 9.0 = 6.1\ &BE = BD = 6.1\ &CE = CF = 5.6 \end{aligned} $$ ## Calculate the side lengths of the triangle $$ \begin{aligned} &AB = 15.1\ &AC = 14.6\ &BC = BE + CE = 6.1 + 5.6 = 11.7 \end{aligned} $$ ## Calculate the perimeter of the triangle $$ \begin{aligned} &P = AB + AC + BC\ &P = 15.1 + 14.6 + 11.7 = 41.4 \end{aligned} $$ Find the perimeter of the triangle. P = 41.4 { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Geometry", "Tangent Segments Theorem" ] }

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Explanation:

Determine the lengths of the tangent segments from each vertex

Calculate the side lengths of the triangle

Calculate the perimeter of the triangle

Answer: