in the diagram below, $\delta jkl$ is an equilateral triangle and $\overline{km} \perp \overline{jl}$.

which statement must be true?

a. $km = 2 \cdot jm$
b. $jk = km$
c. $\delta jkm$ is a 45-45-90 triangle.
d. $\delta jkm$ is a 30-60-90 triangle.

Question

in the diagram below, $\delta jkl$ is an equilateral triangle and $\overline{km} \perp \overline{jl}$.

which statement must be true?

a. $km = 2 \cdot jm$
b. $jk = km$
c. $\delta jkm$ is a 45-45-90 triangle.
d. $\delta jkm$ is a 30-60-90 triangle.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Special Right Triangles", "Triangle Angle Sum Theorem" ], "new_concepts": [ "Equilateral Triangle Properties" ], "current_concepts": [ "Equilateral Triangle Properties", "Special Right Triangles", "Triangle Angle Sum Theorem" ] } ## Identify properties of the equilateral triangle Using the *Triangle Angle Sum Theorem* knowledge point $$ \begin{aligned} &\triangle JKL \text{ is equilateral} \implies m\angle J = m\angle K = m\angle L = 60^\circ\ &JK = KL = LJ \end{aligned} $$ ## Analyze the perpendicular segment Using the *Special Right Triangles* knowledge point $$ \begin{aligned} &\overline{KM} \perp \overline{JL} \implies m\angle KMJ = 90^\circ\ &\triangle JKM \text{ is a right triangle} \end{aligned} $$ ## Determine the angles of the right triangle Using the *Triangle Angle Sum Theorem* knowledge point $$ \begin{aligned} &m\angle JKM = 180^\circ - m\angle KMJ - m\angle J\ &m\angle JKM = 180^\circ - 90^\circ - 60^\circ = 30^\circ \end{aligned} $$ ## Classify the resulting triangle Using the *Special Right Triangles* knowledge point $$ \begin{aligned} &\text{Angles of } \triangle JKM \text{ are } 30^\circ, 60^\circ, \text{ and } 90^\circ\ &\triangle JKM \text{ is a 30-60-90 triangle} \end{aligned} $$ ## Evaluate the given options Using the *Special Right Triangles* knowledge point $$ \begin{aligned} &\text{In a 30-60-90 triangle, } JK = 2 \cdot JM \text{ and } KM = \sqrt{3} \cdot JM\ &\text{Thus, Option A is false, Option B is false, Option C is false, Option D is true} \end{aligned} $$

A. $KM = 2 \cdot JM$

B. $JK = KM$

C. $\triangle JKM$ is a 45-45-90 triangle.

D. $\triangle JKM$ is a 30-60-90 triangle.

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "Special Right Triangles" ] }

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QUESTION IMAGE

Question

Explanation:

Identify properties of the equilateral triangle

Analyze the perpendicular segment

Determine the angles of the right triangle

Classify the resulting triangle

Evaluate the given options

Answer: