Final answer:
The measure of angle Q in triangle QRS can be found using the Law of Cosines. For sides q = 83 inches, r = 92 inches, and s = 50 inches, applying the formula gives Q ≈ cos⁻¹(0.8423), which calculates to approximately 32.7 degrees.
Step-by-step explanation:
To find the measure of angle Q in triangle QRS, with sides q = 83 inches, r = 92 inches, and s = 50 inches, one can use the Law of Cosines. This law states that in any triangle, the square of one side is equal to the sum of the squares of the other two sides minus twice the product of those sides and the cosine of the included angle.
Following this law for angle Q:
c2 = a2 + b2 - 2*a*b*cos(C)
We substitute the lengths: s2 = q2 + r2 - 2*q*r*cos(Q). Solving for cos(Q):
cos(Q) = (q2 + r2 - s2) / (2*q*r)
cos(Q) = (832 + 922 - 502) / (2*83*92)
cos(Q) = (6889 + 8464 - 2500) / (15256)
cos(Q) = (12853) / (15256)
cos(Q) ≈ 0.8423
Finally, to find the measure of angle Q, use the inverse cosine function:
Q ≈ cos-1(0.8423)
Using a calculator, we find that:
Q ≈ 32.7 degrees
Therefore, the measure of angle Q in triangle QRS is approximately 32.7 degrees.