Question

lines cd and de are tangent to circle a, as shown below:

if \\(\widehat{ce}\\) is \\(110^\circ\\), what is the measure of \\(\angle cde\\)?

• \\(55^\circ\\)
• \\(70^\circ\\)
• \\(100^\circ\\)
• \\(125^\circ\\)

Explanation:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
"quick"
],
"needs_drawing": true,
"concepts_used": [
"Circle Geometry"
],
"new_concepts": [
"Tangent-Tangent Angle Theorem",
"Circumscribed Angle"
],
"current_concepts": [
"Circle Geometry",
"Tangent-Tangent Angle Theorem",
"Circumscribed Angle"
]
}
</pre_analysis>

<reasoning>

Determine the measure of the major arc CBE

\[
m(\widehat{CBE}) = 360^\circ - m(\widehat{CE}) = 360^\circ - 110^\circ = 250^\circ
\]

Apply the Tangent-Tangent Angle Theorem

\[
m\angle CDE = \frac{m(\widehat{CBE}) - m(\widehat{CE})}{2}
\]
\[
m\angle CDE = \frac{250^\circ - 110^\circ}{2} = \frac{140^\circ}{2} = 70^\circ
\]
</reasoning>

<answer>
<mcq-option>(A) \(55^\circ\)</mcq-option>
<mcq-correct>(B) \(70^\circ\)</mcq-correct>
<mcq-option>(C) \(100^\circ\)</mcq-option>
<mcq-option>(D) \(125^\circ\)</mcq-option>
</answer>

<plot>
{
"elements": [
{
"type": "point",
"params": [[0, 0]],
"properties": {
"name": "A",
"color": "#583C87",
"withLabel": true
}
},
{
"type": "circle",
"params": [[0, 0], 3],
"properties": {
"strokeColor": "#8C55F2",
"strokeWidth": 2
}
},
{
"type": "point",
"params": [[-1.026, 2.819]],
"properties": {
"name": "C",
"color": "#583C87",
"withLabel": true
}
},
{
"type": "point",
"params": [[1.026, 2.819]],
"properties": {
"name": "E",
"color": "#583C87",
"withLabel": true
}
},
{
"type": "point",
"params": [[0, 8.24]],
"properties": {
"name": "D",
"color": "#583C87",
"withLabel": true
}
},
{
"type": "segment",
"params": [[-1.026, 2.819], [0, 8.24]],
"properties": {
"strokeColor": "#5583F2",
"strokeWidth": 2
}
},
{
"type": "segment",
"params": [[1.026, 2.819], [0, 8.24]],
"properties": {
"strokeColor": "#5583F2",
"strokeWidth": 2
}
},
{
"type": "point",
"params": [[-2.12, -2.12]],
"properties": {
"name": "B",
"color": "#583C87",
"withLabel": true
}
}
],
"timestamps": [0.5, 1.0]
}
</plot>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Geometry",
"Circle Geometry"
]
}
</post_analysis>

Answer:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
"quick"
],
"needs_drawing": true,
"concepts_used": [
"Circle Geometry"
],
"new_concepts": [
"Tangent-Tangent Angle Theorem",
"Circumscribed Angle"
],
"current_concepts": [
"Circle Geometry",
"Tangent-Tangent Angle Theorem",
"Circumscribed Angle"
]
}
</pre_analysis>

<reasoning>

Determine the measure of the major arc CBE

\[
m(\widehat{CBE}) = 360^\circ - m(\widehat{CE}) = 360^\circ - 110^\circ = 250^\circ
\]

Apply the Tangent-Tangent Angle Theorem

\[
m\angle CDE = \frac{m(\widehat{CBE}) - m(\widehat{CE})}{2}
\]
\[
m\angle CDE = \frac{250^\circ - 110^\circ}{2} = \frac{140^\circ}{2} = 70^\circ
\]
</reasoning>

<answer>
<mcq-option>(A) \(55^\circ\)</mcq-option>
<mcq-correct>(B) \(70^\circ\)</mcq-correct>
<mcq-option>(C) \(100^\circ\)</mcq-option>
<mcq-option>(D) \(125^\circ\)</mcq-option>
</answer>

<plot>
{
"elements": [
{
"type": "point",
"params": [[0, 0]],
"properties": {
"name": "A",
"color": "#583C87",
"withLabel": true
}
},
{
"type": "circle",
"params": [[0, 0], 3],
"properties": {
"strokeColor": "#8C55F2",
"strokeWidth": 2
}
},
{
"type": "point",
"params": [[-1.026, 2.819]],
"properties": {
"name": "C",
"color": "#583C87",
"withLabel": true
}
},
{
"type": "point",
"params": [[1.026, 2.819]],
"properties": {
"name": "E",
"color": "#583C87",
"withLabel": true
}
},
{
"type": "point",
"params": [[0, 8.24]],
"properties": {
"name": "D",
"color": "#583C87",
"withLabel": true
}
},
{
"type": "segment",
"params": [[-1.026, 2.819], [0, 8.24]],
"properties": {
"strokeColor": "#5583F2",
"strokeWidth": 2
}
},
{
"type": "segment",
"params": [[1.026, 2.819], [0, 8.24]],
"properties": {
"strokeColor": "#5583F2",
"strokeWidth": 2
}
},
{
"type": "point",
"params": [[-2.12, -2.12]],
"properties": {
"name": "B",
"color": "#583C87",
"withLabel": true
}
}
],
"timestamps": [0.5, 1.0]
}
</plot>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Geometry",
"Circle Geometry"
]
}
</post_analysis>