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

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

Accepted Answer

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

14 units

15 units

16 units

17 units

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "Circle Geometry" ] }

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QUESTION IMAGE

Question

Explanation:

Identify the geometric properties of the tangent line

Determine the lengths of the triangle sides

Apply the Pythagorean theorem to solve for ZY

Answer: