3.4.4 quiz: 30 - 60 - 90 right triangles
in the diagram below, △wxy is an equilateral triangle and $ \overline{xz} \perp \overline{wy} $.
which statement must be true?
a. $ wx = 2 \cdot wz $
b. $ wx = 2 \cdot xz $
c. $ wx = wz $

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# Brief Explanations:
1. Recall properties of equilateral triangles: All sides are equal, and all angles are $60^\circ$. Also, the altitude (here $XZ$) in an equilateral triangle bisects the base ($WY$) and creates two $30 - 60 - 90$ right triangles ($\triangle WXZ$ and $\triangle YXZ$).
2. In a $30 - 60 - 90$ right triangle, the sides are in the ratio $1:\sqrt{3}:2$, where the side opposite $30^\circ$ is the shortest one (let's call it $a$), the side opposite $60^\circ$ is $a\sqrt{3}$, and the hypotenuse is $2a$.
3. In $\triangle WXZ$, $\angle W = 60^\circ$, $\angle XZW = 90^\circ$, so $\angle WXZ = 30^\circ$. The side opposite $30^\circ$ is $WZ$, and the hypotenuse is $WX$.
4. By the $30 - 60 - 90$ triangle ratio, if $WZ = a$ (opposite $30^\circ$), then the hypotenuse $WX = 2a$, which means $WX = 2\cdot WZ$.
5. For option B: $XZ$ is opposite $60^\circ$ in $\triangle WXZ$, so $XZ = WZ\sqrt{3}$, and $WX = 2WZ$, so $WX
eq2\cdot XZ$ (since $XZ = WZ\sqrt{3}$, $2\cdot XZ = 2WZ\sqrt{3}
eq2WZ$ unless $WZ = 0$, which is impossible).
6. For option C: $WX$ is the hypotenuse and $WZ$ is a leg, so $WX>WZ$, so $WX
eq WZ$.

# Answer:
A. $WX = 2\cdot WZ$

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