Combined triangles

Question

Answer 1

Both triangles are 30-60-90 right triangles, where the sides are in ratio 1:sqrt(3):2 for side opposite the 30, 60 and 90 degree angles.

The side opposite the 60 degree angle in the upper triangle is 6, since ratio of side opposite 60 degree angle to side opposite 90 degree angle is sqrt(3)/2, we can find hypotenuse:

sqrt(3) 6

--------- -----

2 hypotenuse

hypotenuse = 4* sqrt(3)

In the lower triangle that hypotenuse is the side opposite the 60 degree angle. x is the side opposite the 90 degree angle:

sqrt(3) = 4 * sqrt(3)

--------- ------------

2 x

x = 8

Answer 2

sin(60)=6/(hypotenuse of upper triangle)=√3/2

so hypotenuse of upper triangle = 4√3 = long right-angle side of lower triangle,

now we have sin(60)=4√3/(hypotenuse of lower triangle which is x)=√3/2

result ==x=8

Answer 3

In what ratio are the sides of a 30-60-90 degree triangle?

Label all sides.