Question

I am trying to finish geometry over the summer.

Answer 1

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
`s` the small side is
`\frac s 2` and the middle side
`h` (which would be an altitude of the equilateral triangle). We write the Pythagorean Theorem:


`(\frac s 2)^2 + h^2 = s^2`


`\frac {s^2}{4} + h^2 = s^2`


`h^2 = \frac 3 4 s^2`


`h = (√(3))/(2) s`

So the sides opposite 30/60/90 respectively are in the ratio of
`\frac 1 2 : (√(3))/(2) : 1` or equivalently
`1 : √(3) : 2`.

We have E=30, F=60, G=90, with respective opposite sides FG:EG:EF in ratio
`1 : √(3) : 2`. From the ratios we get


`(FG)/(1) = (EG)/(√(3)) = (EF)/(2)`

A.
`EG= (√(3))/(2) EF \quad` TRUE

B. FALSE,
`EG=√(3) FG`

C. FALSE,
`EF=2FG`

D. TRUE (see above)

E. TRUE (see above)

F. FALSE (see above)

That looks good.