8. (10 pts) if triangle log is equilateral with $ \overline{lg} = 3x^2 - 174 $ and $ \overline{go} = 82 - x^2 $, solve for $ x $ and find the perimeter of the triangle.

Question

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Equilateral Triangle Properties" ], "new_concepts": [ "Quadratic Equation Solving", "Geometric Perimeter Calculation" ], "current_concepts": [ "Equilateral Triangle Properties", "Quadratic Equation Solving", "Geometric Perimeter Calculation" ] } ## Equate the side lengths Using the *Equilateral Triangle Properties* knowledge point $$ \begin{aligned} &LG = GO \ &3x^2 - 174 = 82 - x^2 \end{aligned} $$ ## Solve for x Using the *Linear Equation Solving* knowledge point $$ \begin{aligned} &4x^2 = 256 \ &x^2 = 64 \ &x = 8 \quad (\text{since } x > 0 \text{ for positive side lengths}) \end{aligned} $$ ## Calculate side length Using the *Equilateral Triangle Properties* knowledge point $$ \begin{aligned} &LG = 3(8)^2 - 174 \ &LG = 3(64) - 174 \ &LG = 192 - 174 = 18 \end{aligned} $$ ## Calculate the perimeter Using the *Geometric Perimeter Calculation* knowledge point $$ \begin{aligned} &P = 3 \times LG \ &P = 3 \times 18 = 54 \end{aligned} $$ $x = 8$, Perimeter = $54$ { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Geometry", "Equilateral Triangle Properties" ] }

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Question

Explanation:

Equate the side lengths

Solve for x

Calculate side length

Calculate the perimeter

Answer: