Question

1. find all the missing side lengths for the following triangles. (3 points)

a.

b.

c.

d.

Explanation:

<pre_analysis>
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"explain_then_answer",
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"needs_drawing": false,
"concepts_used": [
"Trigonometric Ratios",
"Pythagorean Theorem",
"Right Triangle Sides"
],
"new_concepts": [
"Special Right Triangles",
"30-60-90 Triangle Properties",
"45-45-90 Triangle Properties"
],
"current_concepts": [
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"Pythagorean Theorem",
"Right Triangle Sides",
"Special Right Triangles",
"30-60-90 Triangle Properties",
"45-45-90 Triangle Properties"
]
}
</pre_analysis>

<reasoning>

Solve triangle a (30-60-90 triangle)

\[

$$\begin{aligned} &\text{Opposite side } = 2\\ &u = \frac{2}{\sin(30^\circ)} = 4\\ &v = \frac{2}{\tan(30^\circ)} = 2\sqrt{3} \end{aligned}$$

\]

Solve triangle b (30-60-90 triangle)

\[

$$\begin{aligned} &\text{Opposite side to } 60^\circ = 5\sqrt{3}\\ &c = \frac{5\sqrt{3}}{\sin(60^\circ)} = 10\\ &b = \frac{5\sqrt{3}}{\tan(60^\circ)} = 5 \end{aligned}$$

\]

Solve triangles c and d (45-45-90 triangles)

\[

$$\begin{aligned} &\text{For triangle c:}\\ &b = 2\sqrt{2}\\ &a = 2\sqrt{2} \cdot \sqrt{2} = 4\\ &\text{For triangle d:}\\ &m = n = \frac{7}{\sqrt{2}} = \frac{7\sqrt{2}}{2} \end{aligned}$$

\]
</reasoning>

<answer>

Question a

\(u = 4\), \(v = 2\sqrt{3}\)

Question b

\(b = 5\), \(c = 10\)

Question c

\(a = 4\), \(b = 2\sqrt{2}\)

Question d

\(m = \frac{7\sqrt{2}}{2}\), \(n = \frac{7\sqrt{2}}{2}\)
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multi-part",
"knowledge_point": [
"Mathematics",
"Geometry",
"Special Right Triangles"
]
}
</post_analysis>

Answer:

<pre_analysis>
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"question_count": 4,
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"step_cot",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [
"Trigonometric Ratios",
"Pythagorean Theorem",
"Right Triangle Sides"
],
"new_concepts": [
"Special Right Triangles",
"30-60-90 Triangle Properties",
"45-45-90 Triangle Properties"
],
"current_concepts": [
"Trigonometric Ratios",
"Pythagorean Theorem",
"Right Triangle Sides",
"Special Right Triangles",
"30-60-90 Triangle Properties",
"45-45-90 Triangle Properties"
]
}
</pre_analysis>

<reasoning>

Solve triangle a (30-60-90 triangle)

\[

$$\begin{aligned} &\text{Opposite side } = 2\\ &u = \frac{2}{\sin(30^\circ)} = 4\\ &v = \frac{2}{\tan(30^\circ)} = 2\sqrt{3} \end{aligned}$$

\]

Solve triangle b (30-60-90 triangle)

\[

$$\begin{aligned} &\text{Opposite side to } 60^\circ = 5\sqrt{3}\\ &c = \frac{5\sqrt{3}}{\sin(60^\circ)} = 10\\ &b = \frac{5\sqrt{3}}{\tan(60^\circ)} = 5 \end{aligned}$$

\]

Solve triangles c and d (45-45-90 triangles)

\[

$$\begin{aligned} &\text{For triangle c:}\\ &b = 2\sqrt{2}\\ &a = 2\sqrt{2} \cdot \sqrt{2} = 4\\ &\text{For triangle d:}\\ &m = n = \frac{7}{\sqrt{2}} = \frac{7\sqrt{2}}{2} \end{aligned}$$

\]
</reasoning>

<answer>

Question a

\(u = 4\), \(v = 2\sqrt{3}\)

Question b

\(b = 5\), \(c = 10\)

Question c

\(a = 4\), \(b = 2\sqrt{2}\)

Question d

\(m = \frac{7\sqrt{2}}{2}\), \(n = \frac{7\sqrt{2}}{2}\)
</answer>

<post_analysis>
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"question_type": "Multi-part",
"knowledge_point": [
"Mathematics",
"Geometry",
"Special Right Triangles"
]
}
</post_analysis>