select all the correct answers.

which three pairs of side lengths are possible measurements for the triangle?

$bc = 16$, $ac = 32$
$bc = 19\sqrt{3}$, $ac = 38\sqrt{3}$
$ab = 14$, $ac = 28$
$ab = 3$, $bc = 3\sqrt{3}$
$ab = 4\sqrt{3}$, $bc = 4$
$bc = 5\sqrt{2}$, $ac = 10$

Question

select all the correct answers.

which three pairs of side lengths are possible measurements for the triangle?

$bc = 16$, $ac = 32$
  $bc = 19\sqrt{3}$, $ac = 38\sqrt{3}$
  $ab = 14$, $ac = 28$
  $ab = 3$, $bc = 3\sqrt{3}$
  $ab = 4\sqrt{3}$, $bc = 4$
  $bc = 5\sqrt{2}$, $ac = 10$

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": [ "Special Right Triangles" ], "new_concepts": [], "current_concepts": [ "Special Right Triangles", "Trigonometric Ratios" ] } ## Identify triangle properties Using the *Special Right Triangles* knowledge point $$ \begin{aligned} &\angle B = 90^\circ,\quad \angle A = 30^\circ,\quad \angle C = 60^\circ\ &BC = x\ &AC = 2x\ &AB = x\sqrt{3} \end{aligned} $$ ## Evaluate first option Using the *Special Right Triangles* knowledge point $$ \begin{aligned} &BC = 16,\quad AC = 32\ &\frac{AC}{BC} = \frac{32}{16} = 2\ &AC = 2 \cdot BC \quad (\text{Valid}) \end{aligned} $$ ## Evaluate second option Using the *Special Right Triangles* knowledge point $$ \begin{aligned} &BC = 19\sqrt{3},\quad AC = 38\sqrt{3}\ &\frac{AC}{BC} = \frac{38\sqrt{3}}{19\sqrt{3}} = 2\ &AC = 2 \cdot BC \quad (\text{Valid}) \end{aligned} $$ ## Evaluate remaining options Using the *Special Right Triangles* knowledge point $$ \begin{aligned} &AB = 14,\quad AC = 28 \implies \frac{AC}{AB} = 2 e \frac{2}{\sqrt{3}} \quad (\text{Invalid})\ &AB = 3,\quad BC = 3\sqrt{3} \implies \frac{AB}{BC} = \frac{1}{\sqrt{3}} e \sqrt{3} \quad (\text{Invalid})\ &AB = 4\sqrt{3},\quad BC = 4 \implies \frac{AB}{BC} = \sqrt{3} \quad (\text{Valid})\ &BC = 5\sqrt{2},\quad AC = 10 \implies \frac{AC}{BC} = \sqrt{2} e 2 \quad (\text{Invalid}) \end{aligned} $$

[x] $BC = 16, AC = 32$

[x] $BC = 19\sqrt{3}, AC = 38\sqrt{3}$

[ ] $AB = 14, AC = 28$

[ ] $AB = 3, BC = 3\sqrt{3}$

[x] $AB = 4\sqrt{3}, BC = 4$

[ ] $BC = 5\sqrt{2}, AC = 10$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "Special Right Triangles" ] }

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QUESTION IMAGE

Question

Explanation:

Identify triangle properties

Evaluate first option

Evaluate second option

Evaluate remaining options

Answer: