point f is on circle c.
what is the length of line segment gf?
12.5 units
15.0 units
17.5 units
20.0 units

Question

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [ "Tangent Line to Circle", "Pythagorean Theorem", "Circle Radius" ], "current_concepts": [ "Tangent Line to Circle", "Pythagorean Theorem", "Circle Radius" ] } ## Identify the right triangle and given lengths $$ \begin{aligned} &\angle GEC = 90^\circ \quad (\text{tangent } GE \perp \text{radius } EC) \ &GE = 10 \ &EC = 7.5 \quad (\text{radius } r) \end{aligned} $$ ## Calculate the hypotenuse GC $$ \begin{aligned} &GC = \sqrt{GE^2 + EC^2} \ &GC = \sqrt{10^2 + 7.5^2} = \sqrt{100 + 56.25} = \sqrt{156.25} = 12.5 \end{aligned} $$ ## Calculate the length of segment GF $$ \begin{aligned} &CF = r = 7.5 \ &GF = GC + CF = 12.5 + 7.5 = 20.0 \end{aligned} $$

12.5 units

15.0 units

17.5 units

20.0 units

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "Tangent Line to Circle" ] }

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Question

Explanation:

Identify the right triangle and given lengths

Calculate the hypotenuse GC

Calculate the length of segment GF

Answer: