In a [tex]$30^\circ-60^\circ-90^\circ$[/tex] triangle, the properties of the sides are defined as follows:
- The hypotenuse is twice as long as the shorter leg.
- The longer leg is [tex]$\sqrt{3}$[/tex] times as long as the shorter leg.
Let's analyze each statement one-by-one based on these properties.
A. The hypotenuse is [tex]$\sqrt{3}$[/tex] times as long as the longer leg.
This is incorrect. The hypotenuse is not [tex]$\sqrt{3}$[/tex] times the longer leg; the correct relationship is that the hypotenuse is twice as long as the shorter leg.
B. The longer leg is [tex]$\sqrt{3}$[/tex] times as long as the shorter leg.
This is correct. One of the fundamental properties of a [tex]$30^\circ-60^\circ-90^\circ$[/tex] triangle is that the longer leg is [tex]$\sqrt{3}$[/tex] times the length of the shorter leg.
C. The longer leg is twice as long as the shorter leg.
This is incorrect. The relationship between the longer leg and the shorter leg is that the longer leg is [tex]$\sqrt{3}$[/tex] times the shorter leg, not twice the shorter leg.
D. The hypotenuse is [tex]$\sqrt{3}$[/tex] times as long as the shorter leg.
This is incorrect. The hypotenuse is twice as long as the shorter leg, not [tex]$\sqrt{3}$[/tex] times the shorter leg.
E. The hypotenuse is twice as long as the shorter leg.
This is correct. In a [tex]$30^\circ-60^\circ-90^\circ$[/tex] triangle, the hypotenuse is always twice the length of the shorter leg.
F. The hypotenuse is twice as long as the longer leg.
This is incorrect. The hypotenuse is twice as long as the shorter leg, not the longer leg.
Thus, the true statements about a [tex]$30^\circ-60^\circ-90^\circ$[/tex] triangle are:
- B. The longer leg is [tex]$\sqrt{3}$[/tex] times as long as the shorter leg.
- E. The hypotenuse is twice as long as the shorter leg.
So the correct answers are: B and E.
