Given a triangle with angles of [tex]\( 30^\circ \)[/tex] and [tex]\( 60^\circ \)[/tex]:
1. Step 1: Identify the type of triangle.
- The angle sum property of a triangle states that the sum of the interior angles is [tex]\( 180^\circ \)[/tex].
- Given angles are [tex]\( 30^\circ \)[/tex] and [tex]\( 60^\circ \)[/tex].
- Missing angle = [tex]\( 180^\circ - 30^\circ - 60^\circ = 90^\circ \)[/tex].
- Therefore, this is a 30-60-90 triangle.
2. Step 2: Understand the properties of a 30-60-90 triangle.
- In a 30-60-90 triangle, the sides are in a specific ratio:
- The side opposite the [tex]\( 30^\circ \)[/tex] angle is the shortest and denoted as [tex]\( x \)[/tex].
- The side opposite the [tex]\( 60^\circ \)[/tex] angle is [tex]\( x\sqrt{3} \)[/tex].
- The hypotenuse (opposite the [tex]\( 90^\circ \)[/tex] angle) is [tex]\( 2x \)[/tex].
3. Step 3: Analyze the given statements.
- Statement A: "The longest side is [tex]\(\sqrt{3}\)[/tex] times as long as the shortest side."
- The longest side is the hypotenuse which is [tex]\( 2x \)[/tex].
- [tex]\(\sqrt{3} \times \text{shortest side} = \sqrt{3} \times x \neq 2x \)[/tex].
- Hence, this statement is incorrect.
- Statement B: "The longest side is twice as long as the shortest side."
- As established, the longest side (hypotenuse) is [tex]\( 2x \)[/tex].
- [tex]\( 2 \times \text{shortest side} = 2 \times x = 2x \)[/tex].
- Hence, this statement is correct.
- Statement C: "The second-longest side is twice as long as the shortest side."
- The second-longest side is [tex]\( x\sqrt{3} \)[/tex].
- [tex]\( 2 \times \text{shortest side} = 2 \times x \neq x\sqrt{3} \)[/tex].
- Hence, this statement is incorrect.
- Statement D: "Two sides of the triangle have the same length."
- None of the sides in a 30-60-90 triangle are equal.
- Hence, this statement is incorrect.
4. Conclusion:
- Based on the analysis, the correct statement about the triangle is:
- B. The longest side is twice as long as the shortest side.
Therefore, the answer is [tex]\(\boxed{2}\)[/tex].
