anonymous
  • anonymous
An acute triangle has side 8 and 15. How many possible integral lengths are there for the third side?
Mathematics
schrodinger
  • schrodinger
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wolf1728
  • wolf1728
The third side has to be greater than 7, otherwise a triangle can't be formed. The third side has to be less than 17, otherwise it will be a right triangle or an obtuse triangle. So, that leaves us with possible side lengths of 8, 9 10, 11, 12, 13, 14, 15 and 16 We can use the Law of Cosines to determine if Angle B is less than 90 degrees. cos(B) = (8^2 + 8^2 -15^2) / (2*8*8) cos(B) = -0.7578125 Angle B = 139.27 Degrees cos(B) = (9^2 + 8^2 -15^2) / (2*9*8) cos(B) = -.555555555 cos(B) = 123.75 Degrees We continue with this until we have side a = 13 where angle B equals 87.796 So, the possible lengths for side "a" are 13, 14, 15 and 16

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