Final answer:
To arrange the angles of the triangle by size, we can use the Law of Cosines. By applying the equation, we find that the correct arrangement is ∠P < ∠Q < ∠R.
Step-by-step explanation:
The three sides of a triangle are pq=9, qr=8, and pr=18. To arrange the angles by size, we need to use the Law of Cosines. According to the law, in a triangle, the square of a side is equal to the sum of the squares of the other two sides, minus twice the product of the lengths of those two sides multiplied by the cosine of the included angle.
Using the Law of Cosines, we can determine the measures of the angles. Let's denote the angles as ∠P, ∠Q, and ∠R. We can solve for each angle using the equation:
cos ∠P = (qr^2 + pr^2 - pq^2) / (2 * qr * pr) = (8^2 + 18^2 - 9^2) / (2 * 8 * 18) = 185 / 288
cos ∠Q = (pr^2 + pq^2 - qr^2) / (2 * pr * pq) = (18^2 + 9^2 - 8^2) / (2 * 18 * 9) = 185 / 324
cos ∠R = (pq^2 + qr^2 - pr^2) / (2 * pq * qr) = (9^2 + 8^2 - 18^2) / (2 * 9 * 8) = -265 / 288
Since the angles are named in the order ∠P, ∠Q, and ∠R, we can compare the cosine values to determine the size of the angles.
cos ∠P = 185 / 288, cos ∠Q = 185 / 324, cos ∠R = -265 / 288
Since ∠Q's cosine value is the smallest, ∠P < ∠Q < ∠R. Therefore, the correct option is a) ∠P < ∠Q < ∠R.