Great. If you like it there's more geometry to learn than you can in one lifetime, so you're never finished.
Anyway, this is the 30/60/90 right triangle, which you'll soon learn is the biggest cliche in trigonometry. It's a pretty basic object, being half an equilateral triangle. So if the hypotenuse is [tex]s[/tex] the small side is [tex]\dfrac s 2[/tex] and the middle side [tex]h[/tex] (which would be an altitude of the equilateral triangle). We write the Pythagorean Theorem:
[tex](\frac s 2)^2 + h^2 = s^2[/tex]
[tex]\frac {s^2}{4} + h^2 = s^2[/tex]
[tex]h^2 = \frac 3 4 s^2[/tex]
[tex]h = \frac{\sqrt{3}}{2} s[/tex]
So the sides opposite 30/60/90 respectively are in the ratio of [tex]\frac 1 2 : \frac{\sqrt{3}}{2} : 1[/tex] or equivalently [tex]1 : \sqrt{3} : 2[/tex].
We have E=30, F=60, G=90, with respective opposite sides FG:EG:EF in ratio [tex]1 : \sqrt{3} : 2[/tex]. From the ratios we get
[tex]\dfrac{FG}{1} = \dfrac{EG}{\sqrt{3}} = \dfrac{EF}{2}
[/tex]
A. [tex]EG= \dfrac{\sqrt{3}}{2} EF \quad[/tex] TRUE
B. FALSE, [tex]EG=\sqrt{3} FG
[/tex]
C. FALSE, [tex]EF=2FG[/tex]
D. TRUE (see above)
E. TRUE (see above)
F. FALSE (see above)
That looks good.
