Let's identify the true statements about a [tex]\(30^\circ - 60^\circ - 90^\circ\)[/tex] triangle by analyzing its properties.
For such a triangle:
- The side opposite the [tex]\(30^\circ\)[/tex] angle is the shortest, called the shorter leg.
- The side opposite the [tex]\(60^\circ\)[/tex] angle is the longer leg.
- The side opposite the [tex]\(90^\circ\)[/tex] angle is the hypotenuse.
The key properties of a [tex]\(30^\circ - 60^\circ - 90^\circ\)[/tex] triangle are:
1. The hypotenuse is twice the length of the shorter leg.
2. The longer leg is [tex]\(\sqrt{3}\)[/tex] times the length of the shorter leg.
Let's consider each statement one by one:
- Statement A: The hypotenuse is twice as long as the shorter leg.
This statement is true. It's a known property of [tex]\(30^\circ - 60^\circ - 90^\circ\)[/tex] triangles.
- Statement B: The longer leg is twice as long as the shorter leg.
This statement is false. The longer leg is actually [tex]\(\sqrt{3}\)[/tex] times the length of the shorter leg, not twice.
- Statement C: The longer leg is [tex]\(\sqrt{3}\)[/tex] times as long as the shorter leg.
This statement is true. This is another known property of [tex]\(30^\circ - 60^\circ - 90^\circ\)[/tex] triangles.
- Statement D: The hypotenuse is [tex]\(\sqrt{3}\)[/tex] times as long as the longer leg.
This statement is false. The hypotenuse is not [tex]\(\sqrt{3}\)[/tex] times the longer leg. Instead, it is twice the length of the shorter leg. Given that the longer leg is [tex]\(\sqrt{3}\)[/tex] times the shorter leg, the hypotenuse is [tex]\(\frac{2}{\sqrt{3}}\)[/tex] times the longer leg, which simplifies to [tex]\(\frac{2\sqrt{3}}{3}\)[/tex] times the longer leg, not [tex]\(\sqrt{3}\)[/tex].
- Statement E: The hypotenuse is twice as long as the longer leg.
This statement is false. The hypotenuse is twice the length of the shorter leg, and the longer leg is [tex]\(\sqrt{3}\)[/tex] times the shorter leg. So, the hypotenuse is [tex]\(\frac{2}{\sqrt{3}}\)[/tex] times the longer leg, which simplifies to [tex]\(\frac{2\sqrt{3}}{3}\)[/tex], not twice.
- Statement F: The hypotenuse is [tex]\(\sqrt{5}\)[/tex] times as long as the shorter leg.
This statement is false. The hypotenuse is exactly twice the length of the shorter leg, not [tex]\(\sqrt{5}\)[/tex].
Therefore, 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.
C. The longer leg is [tex]\(\sqrt{3}\)[/tex] times as long as the shorter leg.
