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{ "quality": "blurry", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Inscribed Angle Theorem", "Circle Geometry" ], "new_concepts": [ "Secant-Secant Theorem", "Intersecting Secants Theorem" ], "current_concepts": [ "Circle Geometry", "Inscribed Angle Theorem", "Intersecting Secants Theorem" ] } ## Identify the geometric configuration and given values $$ \begin{aligned} &\text{Secant lines intersecting outside a circle at point } P.\ &\text{Secant 1 cuts circle at } A \text{ and } B \text{ (closer to farther from } P\text{): } PA = 4, \quad AB = 5 \implies PB = PA + AB = 9.\ &\text{Secant 2 cuts circle at } C \text{ and } D \text{ (closer to farther from } P\text{): } PC = 3, \quad CD = x \implies PD = PC + CD = 3 + x. \end{aligned} $$ ## Apply the Intersecting Secants Theorem $$ \begin{aligned} &PA \cdot PB = PC \cdot PD\ &4 \cdot 9 = 3 \cdot (3 + x)\ &36 = 9 + 3x \end{aligned} $$ ## Solve for the unknown length x $$ \begin{aligned} &3x = 27\ &x = 9 \end{aligned} $$

(A) 5

(B) 7

(C) 9

(D) 12

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

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Question

Explanation:

Identify the geometric configuration and given values

Apply the Intersecting Secants Theorem

Solve for the unknown length x

Answer: