solve for (x), (y), and (z). (bd = 16sqrt{2})

(x)
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(y)
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(z)
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Question

solve for (x), (y), and (z). (bd = 16sqrt{2})

(x)
(x =)

(y)
(y =)

(z)
(z =)

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Razones trigonométricas", "Resolución de triángulos rectángulos", "Triángulos rectángulos especiales" ], "new_concepts": [], "current_concepts": [ "Razones trigonométricas", "Resolución de triángulos rectángulos", "Triángulos rectángulos especiales", "Teorema de Pitágoras" ] } ## Analizar el triángulo rectángulo ABD Using the *Razones trigonométricas* and *Resolución de triángulos rectángulos* knowledge points $$ \begin{aligned} &\angle A = 90^\circ,\quad \angle ADB = 60^\circ,\quad BD = 16\sqrt{2}\ &x = BD \cdot \cos(60^\circ) = 16\sqrt{2} \cdot \frac{1}{2} = 8\sqrt{2}\ &y = BD \cdot \sin(60^\circ) = 16\sqrt{2} \cdot \frac{\sqrt{3}}{2} = 8\sqrt{6} \end{aligned} $$ ## Analizar el triángulo rectángulo isósceles BCD Using the *Triángulos rectángulos especiales* knowledge point $$ \begin{aligned} &\angle C = 90^\circ,\quad BC = CD = z\ &z^2 + z^2 = BD^2\ &2z^2 = (16\sqrt{2})^2 = 256 \cdot 2 = 512\ &z^2 = 256 \implies z = 16 \end{aligned} $$ $x = 8\sqrt{2}$ $y = 8\sqrt{6}$ $z = 16$ { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Geometry", "Trigonometry" ] }

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Explanation:

Analizar el triángulo rectángulo ABD

Analizar el triángulo rectángulo isósceles BCD

Answer: