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
Step 1: Use the angle sum property of a triangle. In any triangle ABC, the sum of angles is A+B+C = 180^. Therefore, B = 180^ - (A+C). Using the cosine identity (180^ - ) = - , we have: B = (180^ - (A+C)) = -(A+C) Step 2: Substitute this into the given equation. The given equation is ( A + A)/( B) = sqrt(2). Substitute B = -(A+C): ( A + A)/(-(A+C)) = sqrt(2) A + A = -sqrt(2) (A+C) Step 3: Expand (A+C) and rearrange the terms. Using the cosine addition formula (X+Y) = X Y - X Y: A + A = -sqrt(2) ( A C - A C) A + A = -sqrt(2) A C + sqrt(2) A C Rearrange the terms to group A and A: A - sqrt(2) A C + A + sqrt(2) A C = 0 Factor out A and A: A (1 - sqrt(2) C) + A (1 + sqrt(2) C) = 0 Step 4: Determine the values of C and C. For the equation A (1 - sqrt(2) C) + A (1 + sqrt(2) C) = 0 to hold true for any angle A in a triangle, the coefficients of A and A must both be zero, as A and A are linearly independent. Thus, we must have: 1 - sqrt(2) C = 0 (Equation 1) 1 + sqrt(2) C = 0 (Equation 2) From Equation 1: sqrt(2) C = 1 C = (1)/(sqrt(2)) From Equation 2: sqrt(2) C = -1 C = -(1)/(sqrt(2)) Step 5: Find the angle C. We need an angle C such that C = (1)/(sqrt(2)) and C = -(1)/(sqrt(2)). Since C is positive and C is negative, angle C must be in the second quadrant. The reference angle for which = (1)/(sqrt(2)) and = (1)/(sqrt(2)) is 45^. In the second quadrant, the angle is 180^ - 45^ = 135^. Therefore, C = 135^. This value is valid for an angle in a triangle (0^ < C < 180^). Thus, we have proven that C = 135^. The final answer is C = 135^