Question

Math help?

What's the value of x?

Answer 1

`x=4`

Explanation:

In a `30-60-90` special right triangle:
Side opposite to `30` degrees: `x`
Side opposite to `60` degrees: `xsqrt3`
Side opposite to `90` degrees: `2x`

So in this case we are given the side opposite to the `60` degrees, because the hypotenuse is always the opposite side of `90` degrees, and the shortest side is always opposite to the shortest angle so in this case `30` degrees:

`xsqrt3=2sqrt3`

Solve for `x` which is the shortest side:
`x=2`

Now solve for the hypotenuse:
`2x= 2*2= 4`

Now referring to the 45-45-90 triangle with the knowledge that the hypotenuse of the previous triangle was `4`, but the hypotenuse for that triangle is actually a leg of this isosceles right triangle:

An isosceles right triangle, is always a 45-45-90 degrees triangle, therefore:
Side opposite to 90 degrees: `xsqrt2`
Side opposite to 45 degrees: `x`
Side opposite to 45 degrees: `x`

Since `x` is also side opposite to `45` degrees:
`x=4`

Answer 2

From the right angled triangle RST we have

`sin60=(RS)/(RT)=>RT=(2sqrt3)/(sqrt3/2)=>RT=4`

From the right angled triangle RQT we have

`tan45=(RQ)/(RT)=>RQ=RT*tan45=4*1=4`

Hence `x=RQ=4`