Answer:
First option.
Option 5.
Option 6.
Step-by-step explanation:
The formula for a 30-60-90 triangle is this:
1) Side opposite to 30 will be value [tex]a[/tex].
2) Side opposite to 60 will be value [tex]a\sqrt{3}[/tex].
3) Hypotenuse will be [tex]2a[/tex].
So let's look and see:
First option: [tex]AB=4[/tex] and [tex]BC=4\sqrt{3}[/tex]
AB is opposite of the angle with 30 degree measurement.
BC is opposite of the angle with 60 degree measurement.
So [tex]a=4[/tex] here.
So the side opposite of 60 using the formula should be [tex]4 \sqrt{3}[/tex] which it is here.
So first option looks good.
Second option: [tex]BC=2\sqrt{3}[/tex] and [tex]AC=2[/tex].
We aren't given the side opposite to 30.
AC is the hypotenuse so 2a=2 which means the side opposite to 30 is a=2/2=1.
This means using the formula that the side opposite to 60 will be [tex]1\sqrt{3}=\sqrt{3}[/tex] but we don't have that.
So not option 2.
Third option: [tex]AB=3[/tex] and [tex]AC=3\sqrt{3}[/tex]
AB is the side opposite of 30, so we have [tex]a=3[/tex]
AC is the hypotenuse so that side should be [tex]2a=6[/tex] and it isn't.
Option 3 is not working.
Fourth option: [tex]BC=10[/tex] and [tex]AC=4\sqrt{3}[/tex]
So we have that [tex]2a=4\sqrt{3}[/tex] which means [tex]a=2\sqrt{3}[/tex] and so [tex]a\sqrt{3}=2\sqrt{3}\sqrt{3}=2(3)=6[/tex] but that is a contradiction because we have this value should be 10.
Not option 4.
Option 5: [tex]AB=7[/tex] and [tex]AC=14[/tex]
So we have [tex]a=7[/tex] and [tex]2a=14[/tex] so this looks good.
Option 6: [tex]AB=11[/tex] and [tex]BC=11\sqrt{3}[/tex]
[tex]a=11[/tex] so [tex]a\sqrt{3}=11\sqrt{3}[/tex] which is what we have.
Option 6 works.
