SOLUTION: An isoceles triangle has congruent sides that each measure 3 feet and a base angle of measure of x as shown in the sketch. The third side has a whole number length, if x <60 degree

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Question 969688: An isoceles triangle has congruent sides that each measure 3 feet and a base angle of measure of x as shown in the sketch. The third side has a whole number length, if x <60 degrees , what are the possibilities of the perimeter of the triangle?
Found 2 solutions by josgarithmetic, MathTherapy:
Answer by josgarithmetic(39616) About Me  (Show Source):
You can put this solution on YOUR website!
v, the vertex angle.
v%2Bx%2Bx=180
v=180-2x

One objective is to find a formula for the base length.
b, the length of the base.
Law of Cosines:
3%5E2%2B3%5E3-2%2A3%2A3%2Acos%28v%29=b%5E2

18-18%2Acos%28v%29=b%5E2
b%5E2=18-18%2Acos%28180-2x%29-----Think or inspect this angle on The Unit Circle.


.... cos%28180-2x%29=-cos%282x%29.

Revise the equation for b%5E2.
b%5E2=18%2B18%2Acos%282x%29

The isosceles triangle to have any meaning and match the given requirement, 0%3Cx%3C60.

Perimeter:
6%2Bsqrt%2818%2B18%2Acos%280%29%29%3Eperimeter%3E6%2Bsqrt%2818%2B18cos%28120%29%29
6%2Bsqrt%2818%2B18%2A1%29%3Eperimeter%3E6%2Bsqrt%2818%2B18%2A%28-cos%2860%29%29%29, again refer to Unit Circle to see.
6%2Bsqrt%2836%29%3Eperimeter%3E6%2Bsqrt%2818%2B18%2A%281%2F2%29%29
12%3Eperimeter%3E6%2Bsqrt%2827%29
highlight%2812%3Eperimeter%3E6%2B3%2Asqrt%283%29%29

Answer by MathTherapy(10551) About Me  (Show Source):
You can put this solution on YOUR website!
An isoceles triangle has congruent sides that each measure 3 feet and a base angle of measure of x as shown in the sketch. The third side has a whole number length, if x <60 degrees , what are the possibilities of the perimeter of the triangle?
Third side's length is a whole number

Third side's length > 0, but < 6

Third side is the longest side

Third side's length > 3 feet

Therefore, third side can either be 4 or 5

Thus, perimeter is either: highlight_green%28system%2810_or%2C11%29%29 feet