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
Fresh day Luke, let's solve. Here are the solutions for the given triangles using the Sine Rule and the angle sum property of a triangle, with answers rounded to the nearest whole number. The Sine Rule states: (a)/( A) = (b)/( B) = (c)/( C) The sum of angles in a triangle is 180^: A + B + C = 180^ 1) Given: a = 24.5 cm, B = 44^, C = 76^ Step 1: Find angle A. Using the angle sum property: A = 180^ - B - C A = 180^ - 44^ - 76^ A = 180^ - 120^ A = 60^ Step 2: Find side b. Using the Sine Rule: (b)/( B) = (a)/( A) b = (a B)/( A) b = (24.5 44^)/( 60^) b ≈ (24.5 × 0.6947)/(0.8660) b ≈ 19.65 cm Rounding to the nearest whole number: b ≈ 20 cm Step 3: Find side c. Using the Sine Rule: (c)/( C) = (a)/( A) c = (a C)/( A) c = (24.5 76^)/( 60^) c ≈ (24.5 × 0.9703)/(0.8660) c ≈ 27.45 cm Rounding to the nearest whole number: c ≈ 27 cm 2) Given: A = 62^, b = 14 m, C = 70.75^ Step 1: Find angle B. Using the angle sum property: B = 180^ - A - C B = 180^ - 62^ - 70.75^ B = 180^ - 132.75^ B = 47.25^ Rounding to the nearest whole number: B ≈ 47^ Step 2: Find side a. Using the Sine Rule: (a)/( A) = (b)/( B) a = (b A)/( B) a = (14 62^)/( 47.25^) a ≈ (14 × 0.8829)/(0.7342) a ≈ 16.84 m Rounding to the nearest whole number: a ≈ 17 m Step 3: Find side c. Using the Sine Rule: (c)/( C) = (b)/( B) c = (b C)/( B) c = (14 70.75^)/( 47.25^) c ≈ (14 × 0.9442)/(0.7342) c ≈ 18.00 m Rounding to the nearest whole number: c ≈ 18 m 3) Given: A = 31^, B = 65^, c = 16 m Step 1: Find angle C. Using the angle sum property: C = 180^ - A - B C = 180^ - 31^ - 65^ C = 180^ - 96^ C = 84^ Step 2: Find side a. Using the Sine Rule: (a)/( A) = (c)/( C) a = (c A)/( C) a = (16 31^)/( 84^) a ≈ (16 × 0.5150)/(0.9945) a ≈ 8.29 m Rounding to the nearest whole number