To determine which statements about a [tex]\(30^\circ-60^\circ-90^\circ\)[/tex] triangle are true or false, let's analyze each one considering the known properties of such a triangle.
### Properties of a [tex]\(30^\circ-60^\circ-90^\circ\)[/tex] Triangle
In a [tex]\(30^\circ-60^\circ-90^\circ\)[/tex] triangle:
1. The hypotenuse is the longest side.
2. The length of the hypotenuse is twice the length of the shorter leg.
3. The length of the longer leg is [tex]\(\sqrt{3}\)[/tex] times the length of the shorter leg.
Given this information, let's evaluate each statement:
#### Statement A:
The hypotenuse is twice as long as the shorter leg.
This is true. In a [tex]\(30^\circ-60^\circ-90^\circ\)[/tex] triangle, the hypotenuse is indeed twice the length of the shorter leg.
#### Statement B:
The longer leg is [tex]\(\sqrt{3}\)[/tex] times as long as the shorter leg.
This is true. In a [tex]\(30^\circ-60^\circ-90^\circ\)[/tex] triangle, the longer leg is [tex]\(\sqrt{3}\)[/tex] times the length of the shorter leg.
#### Statement C:
The hypotenuse is [tex]\(\sqrt{3}\)[/tex] times as long as the shorter leg.
This is false. The hypotenuse is twice as long as the shorter leg, not [tex]\(\sqrt{3}\)[/tex] times as long.
#### Statement D:
The hypotenuse is [tex]\(\sqrt{3}\)[/tex] times as long as the longer leg.
This is false. The hypotenuse is not [tex]\(\sqrt{3}\)[/tex] times the length of the longer leg.
#### Statement E:
The hypotenuse is twice as long as the longer leg.
This is false. The hypotenuse is not twice the length of the longer leg; it is twice the length of the shorter leg.
#### Statement F:
The longer leg is twice as long as the shorter leg.
This is false. The longer leg is [tex]\(\sqrt{3}\)[/tex] times the length of the shorter leg, not twice.
### Conclusion
The true statements about a [tex]\(30^\circ-60^\circ-90^\circ\)[/tex] triangle are:
- A. 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.
The false statements are:
- C. The hypotenuse is [tex]\(\sqrt{3}\)[/tex] times as long as the shorter leg.
- D. The hypotenuse is [tex]\(\sqrt{3}\)[/tex] times as long as the longer leg.
- E. The hypotenuse is twice as long as the longer leg.
- F. The longer leg is twice as long as the shorter leg.
Hence, the final result is:
[tex]\[
(\text{{True Statements: }} \text{{['A', 'B']}}, \text{{False Statements: }} \text{{['C', 'D', 'E', 'F']}})
\][/tex]
