Question

Find the length of side x in simplest radical form with a rational denominator. The angles are 30-60-90 while the numbers 5 and x.

Answer 1

A 30–60–90 triangle has side ratios:
short leg (opposite 30) = 1·k, long leg (opposite 60) = √3·k, hypotenuse = 2·k.

If the given side 5 is:
1) the hypotenuse: 2k = 5 → k = 5/2.
- short leg = k = 5/2.
- long leg = √3·k = (5√3)/2.
So x = 5/2 or x = (5√3)/2 depending which side x is.

2) the short leg: k = 5.
- long leg = 5√3.
- hypotenuse = 10.
So x = 5√3 or x = 10.

3) the long leg: √3·k = 5 → k = 5/√3 = (5√3)/3 (rationalized).
- short leg = k = (5√3)/3.
- hypotenuse = 2k = (10√3)/3.
So x = (5√3)/3 or x = (10√3)/3.

If you tell me which side is labeled 5 and which is x in the diagram, I’ll give the single required value.