Let's analyze each statement about a 30-60-90 triangle step-by-step.
A 30-60-90 triangle is a special right triangle where the angles are 30 degrees, 60 degrees, and 90 degrees. The sides of such a triangle have a specific ratio:
- The side opposite the 30-degree angle is the shortest side.
- The side opposite the 60-degree angle is the longer leg.
- The side opposite the 90-degree angle is the hypotenuse.
The ratio of the lengths of the sides in a 30-60-90 triangle is 1 : [tex]$\sqrt{3}$[/tex] : 2.
Now, let's evaluate each of the statements one-by-one:
A. The hypotenuse is [tex]$\sqrt{3}$[/tex] times as long as the longer leg.
In a 30-60-90 triangle, the ratio of the hypotenuse to the longer leg is 2 : [tex]$\sqrt{3}$[/tex], not [tex]$\sqrt{3}$[/tex] : [tex]$\sqrt{3}$[/tex]. Hence, this statement is false.
B. The hypotenuse is twice as long as the shorter leg.
Given the ratio of 1 : [tex]$\sqrt{3}$[/tex] : 2, the hypotenuse (2) is indeed twice as long as the shorter leg (1). This statement is true.
C. The hypotenuse is [tex]$\sqrt{3}$[/tex] times as long as the shorter leg.
In a 30-60-90 triangle, the hypotenuse is 2 times the shorter leg, not [tex]$\sqrt{3}$[/tex] times. Hence, this statement is false.
D. The longer leg is twice as long as the shorter leg.
The ratio of the longer leg to the shorter leg in a 30-60-90 triangle is [tex]$\sqrt{3}$[/tex] : 1, not 2:1. Hence, this statement is false.
E. The hypotenuse is twice as long as the longer leg.
In a 30-60-90 triangle, the hypotenuse is 2 units long and the longer leg is [tex]$\sqrt{3}$[/tex] units long, so the hypotenuse is not twice as long as the longer leg. Hence, this statement is false.
F. The longer leg is [tex]$\sqrt{3}$[/tex] times as long as the shorter leg.
This is consistent with the ratio of 1 : [tex]$\sqrt{3}$[/tex] : 2. Therefore, the longer leg (which is [tex]$\sqrt{3}$[/tex]) is [tex]$\sqrt{3}$[/tex] times as long as the shorter leg (which is 1). This statement is true.
Thus, the correct statements about a 30-60-90 triangle are:
B. The hypotenuse is twice as long as the shorter leg.
F. The longer leg is [tex]$\sqrt{3}$[/tex] times as long as the shorter leg.
