SOLUTION: if the measures of angles of an triangle in the ratio of 1:2:3, and if the perimeter of the triangle is 30 + 10 sqrt(3), what is the length of the smallest side ?

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Question 328006: if the measures of angles of an triangle in the ratio of 1:2:3, and if the perimeter of the triangle is 30 + 10 sqrt(3), what is the length of the smallest side ?
Answer by rapaljer(4671) About Me  (Show Source):
You can put this solution on YOUR website!
If the angles are in a ratio of 1:2:3, then let x = smallest angle.

The other angles are 2x and 3x respectively, so
x + 2x + 3x=180
6x=180
x=30
2x=60
3x=90

This is a 30-60-90 triangle, so the sides are in a ratio of 1:2:sqrt%283%29.

Let y= first side
2y= second side (hypotenuse of right triangle!)
y%2Asqrt%283%29= third side

Perimeter=+y%2B+2y+%2B+y%2Asqrt%283%29=30%2B10%2Asqrt%283%29
3y%2B+y%2Asqrt%283%29=+30%2B+10%2Asqrt%283%29

Therefore, y=10.
The smallest side is 10, the second side is 20, and the third side is 10%2Asqrt%283%29. The perimeter checks!!

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Seminole State College of Florida
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