30 30 60 triangle problem help please ?

Question

Answer 1

A 30-60-90 triangle is a special right triangle whose angles are 30º, 60º, and 90º. 

The triangle is special because its side lengths are always in the ratio of 1: √3 : 2. 

Any triangle of the form 30-60-90 can be solved without applying long-step methods such as the Pythagorean Theorem and trigonometric functions.

30-60-90 triangle:

If BC = 4, AB =    AC =

Answer 2

If BC = 4:

4) AB = 4√3

5) AC = 4*2 = 8

If AB = 2√3:

6) BC = AB/√3 = 2√3/√3 = 2

7) AC = 2 BC = 4

Using 'a' as in the attached diagram:

If AB = 9:

8) √3a = 9

a = 9/√3

We prefer to have radical not in the denominator, so multiply by 1 as (√3/√3)

9/√3 (√/3/√3) = 9√3/3

a = BC = 3√3

9) 2a = 2*3√3 = 6√3

AC = 6√3

Answer 3

In a 30-60-90 triangle, the length of the smallest side is half the length of the hypotenuse and the longer length is √3 times the length of the smaller side.

You are told for #4 and #5 that the smallest side is 4, so:

AC = 8

AB = 4√3

You can use the same rules and logic to do the other four questions.