find the length of the indicated segment. assume lines which appear to be tangent are tangent. write your answer as a decimal to the nearest tenth.

the length of the segment is

Question

find the length of the indicated segment. assume lines which appear to be tangent are tangent. write your answer as a decimal to the nearest tenth.

the length of the segment is

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Tangent-Secant Theorem" ], "new_concepts": [], "current_concepts": [ "Tangent-Secant Theorem" ] } ## Identify the given values and the geometric relationship $$ \begin{aligned} &\text{Tangent segment length } (t) = 7.5 \ &\text{External secant segment length } (e) = x \ &\text{Internal secant segment (diameter/chord) } (d) = 5 \ &\text{Total secant segment length } (s) = x + 5 \end{aligned} $$ ## Apply the Tangent-Secant Theorem $$ \begin{aligned} &t^2 = e \cdot s \ &7.5^2 = x(x + 5) \ &56.25 = x^2 + 5x \ &x^2 + 5x - 56.25 = 0 \end{aligned} $$ ## Solve the quadratic equation for x $$ \begin{aligned} &x = \frac{-5 \pm \sqrt{5^2 - 4(1)(-56.25)}}{2} \ &x = \frac{-5 \pm \sqrt{25 + 225}}{2} \ &x = \frac{-5 \pm \sqrt{250}}{2} \ &x = \frac{-5 + 15.811}{2} \approx 5.406 \end{aligned} $$ Find the length of the indicated segment. Assume lines which appear to be tangent are tangent. Write your answer as a decimal to the nearest tenth. The length of the segment is 5.4. { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Geometry", "Tangent-Secant Theorem" ] }

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QUESTION IMAGE

Question

Explanation:

Identify the given values and the geometric relationship

Apply the Tangent-Secant Theorem

Solve the quadratic equation for x

Answer: