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.
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Here's how to solve question 13: Step 1: Determine the sum of the interior angles of a heptagon. A heptagon has n=7 sides. The sum of the interior angles of a polygon is given by the formula (n-2) × 180^. Sum of angles = (7-2) × 180^ = 5 × 180^ = 900^ Step 2: Sum the given interior angles and solve for x. The given angles are x, 2x, 1(1)/(2)x, 2(1)/(2)x, (1)/(2)x, (1)/(4)x, and 1(1)/(4)x. Convert mixed numbers to improper fractions: 1(1)/(2)x = (3)/(2)x, 2(1)/(2)x = (5)/(2)x, 1(1)/(4)x = (5)/(4)x. Sum of angles: x + 2x + (3)/(2)x + (5)/(2)x + (1)/(2)x + (1)/(4)x + (5)/(4)x = 900^ Combine the coefficients of x: (1 + 2 + (3)/(2) + (5)/(2) + (1)/(2) + (1)/(4) + (5)/(4))x = 900^ (3 + (3+5+1)/(2) + (1+5)/(4))x = 900^ (3 + (9)/(2) + (6)/(4))x = 900^ (3 + (9)/(2) + (3)/(2))x = 900^ (3 + (12)/(2))x = 900^ (3 + 6)x = 900^ 9x = 900^ x = (900^)/(9) = 100^ Step 3: Calculate the value of each angle. • x = 100^ • 2x = 2 × 100^ = 200^ • 1(1)/(2)x = (3)/(2) × 100^ = 150^ • 2(1)/(2)x = (5)/(2) × 100^ = 250^ • (1)/(2)x = (1)/(2) × 100^ = 50^ • (1)/(4)x = (1)/(4) × 100^ = 25^ • 1(1)/(4)x = (5)/(4) × 100^ = 125^ a) Calculate the values of the smallest and the biggest angles. The angles are 100^, 200^, 150^, 250^, 50^, 25^, 125^. The smallest angle is 25^. The biggest angle is 250^. Smallest angle = 25^, Biggest angle = 250^ b) Find the difference between the smallest and the biggest angles. Difference = Biggest angle - Smallest angle Difference = 250^ - 25^ = 225^ Difference = 225^ What's next? Send 'em!