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{ "quality": "blurry", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Pythagorean Theorem" ], "new_concepts": [ "Tangent-Radius Theorem" ], "current_concepts": [ "Pythagorean Theorem", "Tangent-Radius Theorem" ] } ## Identify the geometric relationship and given values $$ \begin{aligned} &\text{Radius of the circle: } r = 5.2 \ &\text{Tangent segment: } x \ &\text{Hypotenuse (distance from center to external point): } d = 5.2 + 13 = 18.2 \end{aligned} $$ ## Set up the Pythagorean equation $$ \begin{aligned} &x^2 + r^2 = d^2 \ &x^2 + 5.2^2 = 18.2^2 \end{aligned} $$ ## Solve for $x^2$ and $x$ $$ \begin{aligned} &x^2 + 27.04 = 331.24 \ &x^2 = 304.2 \ &x = \sqrt{304.2} \approx 17.44 \end{aligned} $$ Assume lines which appear to be tangent are tangent. Solve for x. Write your answers as decimals as needed. $x^2 +$ $5.2$$^2 =$ $18.2$$^2$ $x^2 =$ $304.2$ $x =$ $17.4$ { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Geometry", "Tangent-Radius Theorem" ] }

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Explanation:

Identify the geometric relationship and given values

Set up the Pythagorean equation

Solve for \(x^2\) and \(x\)

Answer: