question 8 of 10

a certain triangle has a $30^\circ$ angle and a $60^\circ$ angle. which must be a true statement about the triangle?

a. the second-longest side is twice as long as the shortest side.
b. the longest side is $\sqrt{3}$ times as long as the shortest side.
c. two sides of the triangle have the same length.
d. the longest side is twice as long as the shortest side.

Question

question 8 of 10

a certain triangle has a $30^\circ$ angle and a $60^\circ$ angle. which must be a true statement about the triangle?

a. the second-longest side is twice as long as the shortest side.
b. the longest side is $\sqrt{3}$ times as long as the shortest side.
c. two sides of the triangle have the same length.
d. the longest side is twice as long as the shortest side.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": true, "concepts_used": [ "Triangle Angle Sum Theorem", "30-60-90 Triangle Properties" ], "new_concepts": [], "current_concepts": [ "Triangle Angle Sum Theorem", "30-60-90 Triangle Properties" ] } ## Find the third angle Using the *Triangle Angle Sum Theorem* knowledge point $$ \begin{aligned} \theta &= 180^\circ - (30^\circ + 60^\circ) \ &= 90^\circ \end{aligned} $$ ## Determine side length ratios Using the *30-60-90 Triangle Properties* knowledge point $$ \begin{aligned} &\text{Shortest side (opposite } 30^\circ\text{)} = x \ &\text{Second-longest side (opposite } 60^\circ\text{)} = x\sqrt{3} \ &\text{Longest side (hypotenuse, opposite } 90^\circ\text{)} = 2x \end{aligned} $$ ## Evaluate the given statements Using the *30-60-90 Triangle Properties* knowledge point $$ \begin{aligned} &\text{Ratio of longest side to shortest side} = \frac{2x}{x} = 2 \end{aligned} $$

A. The second-longest side is twice as long as the shortest side.

B. The longest side is $\sqrt{3}$ times as long as the shortest side.

C. Two sides of the triangle have the same length.

D. The longest side is twice as long as the shortest side.

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QUESTION IMAGE

Question

Explanation:

Find the third angle

Determine side length ratios

Evaluate the given statements

Answer: