for the right triangles below, find the exact values of the side lengths $d$ and $b$.

if necessary, write your responses in simplified radical form.

Question

for the right triangles below, find the exact values of the side lengths $d$ and $b$.

if necessary, write your responses in simplified radical form.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [ "Special Right Triangles", "Trigonometric Ratios" ], "current_concepts": [ "Special Right Triangles", "Trigonometric Ratios" ] } ## Solve for d in the 45-45-90 triangle $$ \begin{aligned} &\cos(45^\circ) = \frac{d}{5} \ &d = 5 \cos(45^\circ) = 5 \cdot \frac{\sqrt{2}}{2} = \frac{5\sqrt{2}}{2} \end{aligned} $$ ## Solve for b in the 30-60-90 triangle $$ \begin{aligned} &\tan(60^\circ) = \frac{b}{4} \ &b = 4 \tan(60^\circ) = 4\sqrt{3} \end{aligned} $$ For the right triangles below, find the exact values of the side lengths $d$ and $b$. If necessary, write your responses in simplified radical form. $d =$ $\frac{5\sqrt{2}}{2}$ $b =$ $4\sqrt{3}$ { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Geometry", "Special Right Triangles" ] }

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QUESTION IMAGE

Question

Explanation:

Solve for d in the 45-45-90 triangle

Solve for b in the 30-60-90 triangle

Answer: