To determine which statements about a [tex]\(30-60-90\)[/tex] triangle are true, let's recall the properties of such a triangle. In a [tex]\(30-60-90\)[/tex] triangle:
1. The shorter leg is opposite the [tex]\(30^\circ\)[/tex] angle.
2. The longer leg is opposite the [tex]\(60^\circ\)[/tex] angle.
3. The hypotenuse is opposite the [tex]\(90^\circ\)[/tex] angle.
In this type of triangle:
- The length of the hypotenuse is twice the length of the shorter leg.
- The length of the longer leg is [tex]\(\sqrt{3}\)[/tex] times the length of the shorter leg.
Now let's evaluate each statement:
A. The hypotenuse is twice as long as the shorter leg.
- This is true because, by definition, the hypotenuse is [tex]\(2 \times\)[/tex] the length of the shorter leg.
B. The longer leg is [tex]\(\sqrt{3}\)[/tex] times as long as the shorter leg.
- This is also true, as the longer leg is always [tex]\(\sqrt{3} \times\)[/tex] the length of the shorter leg.
C. The hypotenuse is [tex]\(\sqrt{3}\)[/tex] times as long as the shorter leg.
- This is false. The hypotenuse is [tex]\(2\)[/tex] times the shorter leg, not [tex]\(\sqrt{3}\)[/tex].
D. The hypotenuse is [tex]\(\sqrt{3}\)[/tex] times as long as the longer leg.
- This is false. The ratio of the hypotenuse to the longer leg is [tex]\( \frac{2}{\sqrt{3}} \)[/tex].
E. The hypotenuse is twice as long as the longer leg.
- This is false based on the known properties of the [tex]\(30-60-90\)[/tex] triangle. The hypotenuse is not twice the longer leg.
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 shorter leg, not [tex]\(2\)[/tex] times.
Thus, the statements that are true 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 numerical result reflecting the truth of each statement is:
[tex]\[ (1, 1, 0, 0, 0, 0). \][/tex]
