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: Write the formulas for the sum of interior angles for polygons with n sides and n+2 sides. The sum of the interior angles of a polygon with N sides is given by S_N = (N-2) × 180^. For the polygon with n sides: S_n = (n-2) × 180^ For the polygon with n+2 sides: S_n+2 = ((n+2)-2) × 180^ = n × 180^ Step 2: Set up the ratio of the sums of interior angles. The problem states that the ratio of these sums is 3:4. (S_n)/(S_n+2) = (3)/(4) Substitute the expressions for S_n and S_n+2: ((n-2) × 180^)/(n × 180^) = (3)/(4) Step 3: Solve the equation for n. Cancel out 180^ from the numerator and denominator: (n-2)/(n) = (3)/(4) Cross-multiply: 4(n-2) = 3n Distribute the 4: 4n - 8 = 3n Subtract 3n from both sides: n - 8 = 0 Add 8 to both sides: n = 8 Step 4: Calculate the size of each exterior angle of the polygon with n sides. The polygon with n sides has n=8 sides. For a regular polygon, the size of each exterior angle is given by the formula (360^)/(N), where N is the number of sides. Exterior angle = (360^)/(n) Substitute n=8: Exterior angle = (360^)/(8) Exterior angle = 45^ The size of each exterior angle of the polygon with n sides is 45^ Send me the next one 📸