A triangle includes the following measurements: angle A=17° , side a=6 in. , and side b=9 in. Drawing the triangles, what are two possible lengths for the third side? Enter the shorter possible length first and round your answers to the nearest whole number.
To determine the length of the third side of the triangle, we can use the Law of Cosines:
c^2 = a^2 + b^2 - 2ab*cos(A)
Plugging in the given values, we have:
c^2 = 6^2 + 9^2 - 2(6)(9)*cos(17°)
Evaluating this expression, we find:
c^2 ≈ 36 + 81 - 108*cos(17°)
c^2 ≈ 36 + 81 - 99.6
c^2 ≈ 17.4
Taking the square root of both sides, we have:
c ≈ √17.4
c ≈ 4.17
Therefore, one possible length for the third side of the triangle is approximately 4.17 inches.
Alternatively, since the cosine function is periodic with period 360°, we can consider the supplementary angle to 17°, which is 180° - 17° = 163°. So, we can also solve for the third side using this angle:
c^2 = 6^2 + 9^2 - 2(6)(9)*cos(163°)
Evaluating this expression, we find:
c^2 ≈ 36 + 81 - (-99.6)
c^2 ≈ 216.6
Taking the square root of both sides, we have:
c ≈ √216.6
c ≈ 14.72
Therefore, another possible length for the third side of the triangle is approximately 14.72 inches.
In summary, the two possible lengths for the third side of the triangle are approximately 4.17 inches and 14.72 inches.