what is the length of line segment lj?

9 units
12 units
15 units
18 units

Question

what is the length of line segment lj?

9 units
12 units
15 units
18 units

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Triangle Similarity", "Geometric Similarity" ], "new_concepts": [ "Geometric Mean Theorem", "Right Triangle Altitude" ], "current_concepts": [ "Triangle Similarity", "Geometric Mean Theorem", "Right Triangle Altitude" ] } ## Identify the geometric relationships in the right triangle $$ \begin{aligned} &\angle LKJ = 90^\circ, \quad KM \perp LJ \ &\triangle LMK \sim \triangle KMJ \sim \triangle LKJ \end{aligned} $$ ## Apply the Geometric Mean Theorem for the altitude $$ \begin{aligned} &KM^2 = LM \cdot MJ \ &6^2 = LM \cdot 3 \ &36 = 3 \cdot LM \ &LM = 12 \end{aligned} $$ ## Calculate the total length of segment LJ $$ \begin{aligned} &LJ = LM + MJ \ &LJ = 12 + 3 \ &LJ = 15 \end{aligned} $$

9 units

12 units

15 units

18 units

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "Geometric Mean Theorem" ] }

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QUESTION IMAGE

Question

Explanation:

Identify the geometric relationships in the right triangle

Apply the Geometric Mean Theorem for the altitude

Calculate the total length of segment LJ

Answer: