Question

Find the length of side x in simplest radical form with a rational denominator. The angles are 30-60-90 while the numbers 4 and x.

Answer 1

A 30-60-90 triangle has side ratios (short leg) : (long leg) : (hypotenuse) = 1 : √3 : 2. Let the short leg = s, then long = s√3 and hypotenuse = 2s.

Three possible placements of the given 4:

1. If 4 is the hypotenuse: 2s = 4 → s = 2.
- short leg = 2
- long leg = 2√3
So x = 2 or x = 2√3 depending which side x labels.

2. If 4 is the short leg: s = 4.
- long leg = 4√3
- hypotenuse = 8
So x = 4√3 or x = 8 as applicable.

3. If 4 is the long leg: s√3 = 4 → s = 4/√3 = (4√3)/3 (rationalized).
- short leg = 4√3/3
- hypotenuse = 2s = 8/√3 = (8√3)/3 (rationalized).
So x = 4√3/3 or x = (8√3)/3 as applicable.

Tell me which side in your diagram is labeled 4 (short, long, or hypotenuse) and which side is x, and I’ll give the single correct value.