To determine which statements about a [tex]$30-60-90$[/tex] triangle are true, we first need to understand the specific properties of such a triangle. In a [tex]$30-60-90$[/tex] triangle, the sides have a distinct ratio. Specifically:
- The shorter leg (opposite the [tex]$30^\circ$[/tex] angle) can be represented as [tex]$x$[/tex].
- The longer leg (opposite the [tex]$60^\circ$[/tex] angle) is [tex]$\sqrt{3} \cdot x$[/tex].
- The hypotenuse (opposite the [tex]$90^\circ$[/tex] angle) is [tex]$2x$[/tex].
Now, we'll evaluate each option based on these properties:
A. The longer leg is twice as long as the shorter leg.
[tex]\[ \text{False: The longer leg is } \sqrt{3} \text{ times as long as the shorter leg.} \][/tex]
B. The hypotenuse is twice as long as the longer leg.
[tex]\[ \text{False: The hypotenuse is twice the shorter leg, not the longer leg.} \][/tex]
C. The hypotenuse is twice as long as the shorter leg.
[tex]\[ \text{True: The hypotenuse is indeed twice as long as the shorter leg.} \][/tex]
D. The longer leg is [tex]$\sqrt{3}$[/tex] times as long as the shorter leg.
[tex]\[ \text{True: The longer leg is } \sqrt{3} \text{ times as long as the shorter leg.} \][/tex]
E. The hypotenuse is [tex]$\sqrt{3}$[/tex] times as long as the longer leg.
[tex]\[ \text{False: The hypotenuse is } 2x \text{, while the longer leg is } \sqrt{3} \cdot x\text{.} \][/tex]
[tex]\[ 2x \neq \sqrt{3} \cdot (\sqrt{3} \cdot x) = 3x \][/tex]
F. The hypotenuse is [tex]$\sqrt{3}$[/tex] times as long as the shorter leg.
[tex]\[ \text{False: The hypotenuse is twice the shorter leg, not } \sqrt{3} \text{ times.} \][/tex]
So, the correct evaluations are as follows:
A. False
B. False
C. True
D. True
E. False
F. False
The true statements about a [tex]$30-60-90$[/tex] triangle are:
[tex]\[ \text{C. The hypotenuse is twice as long as the shorter leg.} \][/tex]
[tex]\[ \text{D. The longer leg is } \sqrt{3} \text{ times as long as the shorter leg.} \][/tex]
