Question

what must be the length of \\(\overline{zy}\\) in order for \\(\overline{zy}\\) to be tangent to circle x at point y?

14 units
15 units
16 units
17 units

Explanation:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [
"Circle Geometry"
],
"new_concepts": [
"Tangent Line to Circle",
"Pythagorean Theorem"
],
"current_concepts": [
"Circle Geometry",
"Tangent Line to Circle",
"Pythagorean Theorem"
]
}
</pre_analysis>

<reasoning>

Identify the geometric properties of the tangent line

\[

$$\begin{aligned} &\text{For } \overline{ZY} \text{ to be tangent to circle } X \text{ at point } Y, \text{ the radius } \overline{XY} \text{ must be perpendicular to } \overline{ZY}.\\ &\angle XYZ = 90^\circ\\ &\triangle XYZ \text{ is a right triangle with hypotenuse } XZ. \end{aligned}$$

\]

Determine the lengths of the triangle sides

\[

$$\begin{aligned} &XY = 8 \quad (\text{radius of circle } X)\\ &XZ = 8 + 9 = 17 \quad (\text{radius } + \text{external segment}) \end{aligned}$$

\]

Apply the Pythagorean theorem to solve for ZY

\[

$$\begin{aligned} &XY^2 + ZY^2 = XZ^2\\ &8^2 + ZY^2 = 17^2\\ &64 + ZY^2 = 289\\ &ZY^2 = 225\\ &ZY = 15 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>14 units</mcq-option>
<mcq-correct>15 units</mcq-correct>
<mcq-option>16 units</mcq-option>
<mcq-option>17 units</mcq-option>
</answer>

<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": false,
"concepts_used": [
"Circle Geometry"
],
"new_concepts": [
"Tangent Line to Circle",
"Pythagorean Theorem"
],
"current_concepts": [
"Circle Geometry",
"Tangent Line to Circle",
"Pythagorean Theorem"
]
}
</pre_analysis>

<reasoning>

Identify the geometric properties of the tangent line

\[

$$\begin{aligned} &\text{For } \overline{ZY} \text{ to be tangent to circle } X \text{ at point } Y, \text{ the radius } \overline{XY} \text{ must be perpendicular to } \overline{ZY}.\\ &\angle XYZ = 90^\circ\\ &\triangle XYZ \text{ is a right triangle with hypotenuse } XZ. \end{aligned}$$

\]

Determine the lengths of the triangle sides

\[

$$\begin{aligned} &XY = 8 \quad (\text{radius of circle } X)\\ &XZ = 8 + 9 = 17 \quad (\text{radius } + \text{external segment}) \end{aligned}$$

\]

Apply the Pythagorean theorem to solve for ZY

\[

$$\begin{aligned} &XY^2 + ZY^2 = XZ^2\\ &8^2 + ZY^2 = 17^2\\ &64 + ZY^2 = 289\\ &ZY^2 = 225\\ &ZY = 15 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>14 units</mcq-option>
<mcq-correct>15 units</mcq-correct>
<mcq-option>16 units</mcq-option>
<mcq-option>17 units</mcq-option>
</answer>

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