Which of the following are true statements about a [tex]$30$[/tex]-[tex]$60$[/tex]-[tex]$90$[/tex] triangle? Check all that apply.

A. The hypotenuse is [tex]$\sqrt{3}$[/tex] times as long as the shorter leg.
B. The longer leg is [tex]$\sqrt{3}$[/tex] times as long as the shorter leg.
C. The hypotenuse is [tex]$\sqrt{3}$[/tex] times as long as the longer leg.
D. The longer leg is twice as long as the shorter leg.
E. The hypotenuse is twice as long as the shorter leg.
F. The hypotenuse is twice as long as the longer leg.



Answer :

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.