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
To calculate the area of ABC, we first need to find angle A. We can do this by first finding angle C using the Law of Sines. Step 1: Use the Law of Sines to find C. The Law of Sines states that for a triangle with sides a, b, c and opposite angles A, B, C: (a)/( A) = (b)/( B) = (c)/( C) In ABC, we have side AB = 7 cm (which is c), side AC = 9 cm (which is b), and angle B = 38^. We can set up the ratio to find C: (AC)/( B) = (AB)/( C) Substitute the given values: (9)/( 38^) = (7)/( C) Rearrange to solve for C: C = (7 × 38^)/(9) C = (7 × 0.61566)/(9) C ≈ 0.47885 Step 2: Calculate angle C and check for ambiguous case. C = (0.47885) C ≈ 28.6^ We must check for a second possible angle for C (ambiguous case): 180^ - 28.6^ = 151.4^. If C = 151.4^, then B + C = 38^ + 151.4^ = 189.4^. This sum is greater than 180^, which is not possible for a triangle. Therefore, C ≈ 28.6^ is the only valid angle. Step 3: Calculate angle A. The sum of angles in a triangle is 180^: A = 180^ - B - C A = 180^ - 38^ - 28.6^ A = 180^ - 66.6^ A = 113.4^ Step 4: Calculate the area of ABC. The area of a triangle can be calculated using the formula: Area = (1)/(2)ab C Using sides AB and AC and the included angle A: Area = (1)/(2) × AB × AC × A Area = (1)/(2) × 7 cm × 9 cm × 113.4^ Area = (1)/(2) × 63 cm^2 × 0.9178 Area = 31.5 cm^2 × 0.9178 Area ≈ 28.8967 cm^2 Rounding to one decimal digit: Area ≈ 28.9 cm^2 The area of ABC is 28.9 cm^2. That's 2 down. 3 left today — send the next one.