find the value of each variable.
state your solution in simplest radical form where $x = a\sqrt{b}$, $y = c\sqrt{d}$, and $z = e\sqrt{f}$, where a, c, and e are coefficients and b, d, and f are radicands. use na when necessary.
$a =$
$b =$
$c =$
$d =$
$e =$
$f =$

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{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Geometric Mean", "Right Triangle Similarity", "Radical Simplification" ], "new_concepts": [], "current_concepts": [ "Geometric Mean", "Right Triangle Similarity", "Radical Simplification", "Pythagorean Theorem" ] } ## Solve for x using geometric mean theorem $$ \begin{aligned} 6^2 &= x \cdot 12 \ 36 &= 12x \ x &= 3 \end{aligned} $$ ## Solve for y using geometric mean theorem $$ \begin{aligned} y^2 &= x \cdot (12 - x) \ y^2 &= 3 \cdot (12 - 3) \ y^2 &= 3 \cdot 9 = 27 \ y &= \sqrt{27} = 3\sqrt{3} \end{aligned} $$ ## Solve for z using geometric mean theorem $$ \begin{aligned} z^2 &= (12 - x) \cdot 12 \ z^2 &= 9 \cdot 12 = 108 \ z &= \sqrt{108} = 6\sqrt{3} \end{aligned} $$ Find the value of each variable. State your solution in simplest radical form where $x = a\sqrt{b}$, $y = c\sqrt{d}$, and $z = e\sqrt{f}$, where a, c, and e are coefficients and b, d, and f are radicands. Use NA when necessary. $a =$ 3 $b =$ NA $c =$ 3 $d =$ 3 $e =$ 6 $f =$ 3 { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Geometry", "Geometric Mean in Right Triangles" ] }

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Solve for x using geometric mean theorem

Solve for y using geometric mean theorem

Solve for z using geometric mean theorem

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