In a \(30^\circ\)-\(60^\circ\)-\(90^\circ\) triangle, the sides have a specific ratio. Let's label the sides as follows:
- The side opposite the \(30^\circ\) angle is the shorter leg.
- The side opposite the \(60^\circ\) angle is the longer leg.
- The side opposite the \(90^\circ\) angle is the hypotenuse.
The ratio of the sides in a \(30^\circ\)-\(60^\circ\)-\(90^\circ\) triangle is:
- The hypotenuse is twice as long as the shorter leg.
- The longer leg is \(\sqrt{3}\) times as long as the shorter leg.
Given these ratios, we can now analyze the statements:
A. The hypotenuse is \(\sqrt{3}\) times as long as the shorter leg.
- This is not true. The hypotenuse is actually twice as long as the shorter leg.
B. The longer leg is \(\sqrt{3}\) times as long as the shorter leg.
- This is true, according to the properties of a \(30^\circ\)-\(60^\circ\)-\(90^\circ\) triangle.
C. The hypotenuse is \(\sqrt{3}\) times as long as the longer leg.
- This is not true. Instead, the hypotenuse is \( \frac{2}{\sqrt{3}} \) or \(\frac{2}{\sqrt{3}}\) times the length of the longer leg.
D. The longer leg is twice as long as the shorter leg.
- This is not true. The longer leg is \(\sqrt{3}\) times as long as the shorter leg, not twice as long.
E. The hypotenuse is twice as long as the shorter leg.
- This is true, according to the properties of a \(30^\circ\)-\(60^\circ\)-\(90^\circ\) triangle.
F. The hypotenuse is twice as long as the longer leg.
- This is not true. The hypotenuse is twice as long as the shorter leg, but not the longer leg.
Thus, the true statements about a \(30^\circ\)-\(60^\circ\)-\(90^\circ\) triangle are:
B. The longer leg is \(\sqrt{3}\) times as long as the shorter leg.
E. The hypotenuse is twice as long as the shorter leg.
So, the correct selection is B and E.
