This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
a) To determine the other angles of ABC, we will use the Sine Rule. Step 1: Apply the Sine Rule to find angle C. The Sine Rule states that for a triangle with sides a, b, c and opposite angles A, B, C: (a)/( A) = (b)/( B) = (c)/( C) Given A = 40^, BC = 12 (side a), and AB = 15 (side c). We want to find C. (BC)/( A) = (AB)/( C) (12)/( 40^) = (15)/( C) C = (15 40^)/(12) C = (15 × 0.6427876)/(12) C ≈ 0.8034845 Since C is acute, we take the principal value: C = (0.8034845) C ≈ 53.46^ Step 2: Find angle B using the sum of angles in a triangle. The sum of angles in a triangle is 180^. A + B + C = 180^ 40^ + B + 53.46^ = 180^ B = 180^ - 40^ - 53.46^ B = 180^ - 93.46^ B ≈ 86.54^ The other angles of the triangle are C ≈ 53.46^, B ≈ 86.54^. b) To calculate the lengths of the sides of PQR, we will first find the missing angle and then use the Sine Rule. Step 1: Find angle P using the sum of angles in a triangle. Given Q = 110^ and R = 40^. P + Q + R = 180^ P + 110^ + 40^ = 180^ P = 180^ - 110^ - 40^ P = 180^ - 150^ P = 30^ Step 2: Apply the Sine Rule to find side QR. Given PQ = 13. We want to find QR (opposite P) and PR (opposite Q). (PQ)/( R) = (QR)/( P) (13)/( 40^) = (QR)/( 30^) QR = (13 30^)/( 40^) QR = (13 × 0.5)/(0.6427876) QR = (6.5)/(0.6427876) QR ≈ 10.12 Step 3: Apply the Sine Rule to find side PR. (PQ)/( R) = (PR)/( Q) (13)/( 40^) = (PR)/( 110^) PR = (13 110^)/( 40^) PR = (13 × 0.9396926)/(0.6427876) PR = (12.2160038)/(0.6427876) PR ≈ 19.00 The lengths of the sides are QR ≈ 10.12, PR ≈ 19.00. Drop the next question 📸