The outside arcs for a circle enveloping a triangle will always equal 360.
Set up an equation that has all arcs equal 360:
[tex](3x + 30) + (5x + 10) + (2x + 20) = 360[/tex]
Combine like terms:
[tex]3x + 5x + 2x = 10x[/tex]
[tex]30 + 10 + 20 = 60[/tex]
[tex]10x + 60 = 360[/tex]
Subtract 60 from both sides:
[tex]10x = 300[/tex]
Divide both sides by 10 to get x by itself:
[tex]x = 30[/tex]
We can now find the values of the arcs:
[tex]3(30) + 30 = 90 + 30 = 120[/tex]
[tex]5(30) + 10 = 150 + 10 = 160[/tex]
[tex]2(30) + 20 = 60 + 20 = 80[/tex]
All three arc lengths are different, which means that the triangle's side lengths are all different. This makes the triangle a scalene triangle.
