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.(1 point)
To find the two possible lengths of the third side, we can use the Law of Sines. The Law of Sines states that in a triangle, the ratio of the length of a side to the sine of the opposite angle is constant.
Since we know angle A and side a, we can use the Law of Sines to find the two possible lengths of side c. The Law of Sines equation is:
sin(A) / a = sin(C) / c
Substituting the given values:
sin(17°) / 6 = sin(C) / c
To solve for c, we can use the equation:
c = (6 * sin(C)) / sin(17°)
Using a calculator, we find:
c ≈ 18.96
Therefore, the first possible length for the third side is approximately 19 in.
To find the second possible length, we need to consider the triangle formed by angle A, side a, and side b. In this case, angle C will be greater than 90 degrees, which means there will be no solution using the Law of Sines. Therefore, there is no second possible length for the third side.
Thus, the shorter possible length for the third side is 19 in.