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{ "quality": "blurry", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "tutor" ], "needs_drawing": true, "concepts_used": [], "new_concepts": [ "Tangent-Secant Theorem", "Circle Geometry" ], "current_concepts": [ "Tangent-Secant Theorem", "Circle Geometry" ] } ## Identify the given values We are given a circle $T$ with: - Tangent segment $\overline{AS}$ - Secant segment $\overline{ELS}$ - External secant segment $SL = 8$ - Internal secant segment (chord) $LE = 10$ ## State the applicable theorem The **Tangent-Secant Theorem** states that if a tangent segment and a secant segment are drawn to a circle from an exterior point, then the square of the length of the tangent segment is equal to the product of the lengths of the external secant segment and the entire secant segment: $$ AS^2 = SL \cdot SE $$ ## Calculate the entire secant length The entire secant segment $SE$ is the sum of the external part $SL$ and the internal part $LE$: $$ SE = SL + LE = 8 + 10 = 18 $$ ## Solve for the tangent length Substitute the values into the theorem formula: $$ AS^2 = 8 \cdot 18 $$ $$ AS^2 = 144 $$ $$ AS = \sqrt{144} = 12 $$ ## Match with the given options The calculated length of $\overline{AS}$ is $12$, which corresponds to option (4).

(1) $\sqrt{18}$

(2) $\sqrt{80}$

(3) 9

(4) 12

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Explanation:

Identify the given values

State the applicable theorem

Calculate the entire secant length

Solve for the tangent length

Match with the given options

Answer: