SOLUTION: If the area of traingle is 80 sq unit.the longest side is 20 unit and other side is 10 unit then the third side is what??

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Question 1073094: If the area of traingle is 80 sq unit.the longest side is 20 unit and other side is 10 unit then the third side is what??
Answer by ikleyn(52864) About Me  (Show Source):
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If the area of traingle is 80 sq unit.the longest side is 20 unit and other side is 10 unit then the third side is what??
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The area of a triangle (of any triangle) is half the product of two sides lengths and the sine of the angle between them:

S = %281%2F2%29%2Aa%2Ab%2Asin%28alpha%29.


Substituting the given data, you obtain sin%28alpha%29 = %282%2AS%29%2F%28a%2Ab%29 = %282%2A80%29%2F%2820%2A10%29 = 160%2F200 = 4%2F5.


Then cos%28alpha%29 = +/- sqrt%281-sin%5E2%28alpha%29%29 = +/- 3%2F5.


cos%28alpha%29 can not be negative under the given condition.

Indeed, otherwise the angle alpha is obtuse and our 20 unit side is not the longest side.

So, cos%28alpha%29 = 3%2F5.

Now apply the Cosine Law theorem. You will get for the length of the third side

c%5E2 = a%5E2+%2B+b%5E2+-+2ab%2Acos%28alpha%29 = 10%5E2+%2B+20%5E2+-+2%2A10%2A20%2A%283%2F5%29 = 500+-+2%2A10%2A4%2A3 = 500+-+240 = 260.

Thus  c = sqrt%28260%29 = 16.125 units (approximately).

Solved.