7. determine a radical expression for the length of each of the missing sides.

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Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Simplifying Radicals", "Combining Like Radicals" ], "new_concepts": [ "Geometric Perimeter with Radicals" ], "current_concepts": [ "Simplifying Radicals", "Combining Like Radicals", "Geometric Perimeter with Radicals" ] } ## Identify the given side lengths Using the *Simplifying Radicals* knowledge point $$ \begin{aligned} &\text{Left vertical side } (H) = 2\sqrt{320} - 3\sqrt{24}\ &\text{Bottom horizontal side } (W) = \sqrt{2000} + \sqrt{6}\ &\text{Rightmost vertical side } (v) = 4\sqrt{44} - \sqrt{117}\ &\text{Inner horizontal side } (h) = 5\sqrt{99} - \sqrt{208} \end{aligned} $$ ## Simplify each radical expression Using the *Simplifying Radicals* knowledge point $$ \begin{aligned} &H = 2\sqrt{64 \times 5} - 3\sqrt{4 \times 6} = 16\sqrt{5} - 6\sqrt{6}\ &W = \sqrt{400 \times 5} + \sqrt{6} = 20\sqrt{5} + \sqrt{6}\ &v = 4\sqrt{4 \times 11} - \sqrt{9 \times 13} = 8\sqrt{11} - 3\sqrt{13}\ &h = 5\sqrt{9 \times 11} - \sqrt{16 \times 13} = 15\sqrt{11} - 4\sqrt{13} \end{aligned} $$ ## Set up equations for missing sides Using the *Combining Like Radicals* knowledge point $$ \begin{aligned} &x = H - v = (16\sqrt{5} - 6\sqrt{6}) - (8\sqrt{11} - 3\sqrt{13})\ &y = W - h = (20\sqrt{5} + \sqrt{6}) - (15\sqrt{11} - 4\sqrt{13}) \end{aligned} $$ ## Calculate the missing side lengths Using the *Combining Like Radicals* knowledge point $$ \begin{aligned} &x = 16\sqrt{5} - 6\sqrt{6} - 8\sqrt{11} + 3\sqrt{13}\ &y = 20\sqrt{5} + \sqrt{6} - 15\sqrt{11} + 4\sqrt{13} \end{aligned} $$

(A) $x = 16\sqrt{5} - 6\sqrt{6} - 8\sqrt{11} + 3\sqrt{13}$

(B) $y = 20\sqrt{5} + \sqrt{6} - 15\sqrt{11} + 4\sqrt{13}$

(C) Both: $x = 16\sqrt{5} - 6\sqrt{6} - 8\sqrt{11} + 3\sqrt{13}$ and $y = 20\sqrt{5} + \sqrt{6} - 15\sqrt{11} + 4\sqrt{13}$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Simplifying Radicals" ] }

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Question

Explanation:

Identify the given side lengths

Simplify each radical expression

Set up equations for missing sides

Calculate the missing side lengths

Answer: