A 30-60-90 triangle is a special type of right triangle that has specific relationships between its sides. Let's break down each statement to determine which ones are true.
In a 30-60-90 triangle, the relationships between the sides are:
- 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.
Now, let's evaluate each statement given this information:
A. The hypotenuse is twice as long as the longer leg.
- The hypotenuse is twice as long as the shorter leg, not the longer leg. Therefore, this statement is false.
B. The longer leg is [tex]\(\sqrt{3}\)[/tex] times as long as the shorter leg.
- This is a correct relationship in a 30-60-90 triangle. Therefore, this statement is true.
C. The hypotenuse is [tex]\(\sqrt{3}\)[/tex] times as long as the shorter leg.
- The hypotenuse is twice as long as the shorter leg, not [tex]\(\sqrt{3}\)[/tex] times. Therefore, this statement is false.
D. The longer leg is twice as long as the shorter leg.
- The longer leg is [tex]\(\sqrt{3}\)[/tex] times as long as the shorter leg, not twice as long. Therefore, this statement is false.
E. The hypotenuse is [tex]\(\sqrt{3}\)[/tex] times as long as the longer leg.
- There is no such relationship in a 30-60-90 triangle. The hypotenuse is not [tex]\(\sqrt{3}\)[/tex] times the longer leg. Therefore, this statement is false.
F. The hypotenuse is twice as long as the shorter leg.
- This is a correct relationship in a 30-60-90 triangle. Therefore, this statement is true.
Summarizing the evaluation, the true statements are:
- B. The longer leg is [tex]\(\sqrt{3}\)[/tex] times as long as the shorter leg.
- F. The hypotenuse is twice as long as the shorter leg.
