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 are the solutions to the problems. 47. Step 1: Recall the formula for the sum of interior angles of a polygon. The sum of the interior angles of a polygon with n sides is given by the formula: S = (n-2) × 180^ Step 2: Set up the equation using the given sum. We are given that the sum of the interior angles is 1080^. 1080 = (n-2) × 180 Step 3: Solve for n. Divide both sides by 180: (1080)/(180) = n-2 6 = n-2 Add 2 to both sides: n = 6+2 n = 8 Step 4: Identify the polygon with n=8 sides. A polygon with 8 sides is an octagon. The correct option is A. The name of the polygon is an Octagon. 48. Step 1: Define the angle and its supplement. Let the angle be . The supplement of the angle is 180^ - . Step 2: Set up the equation based on the given condition. The problem states that the supplement of an angle is twice as large as the angle. 180^ - = 2 Step 3: Solve for . Add to both sides of the equation: 180^ = 2 + 180^ = 3 Divide by 3: = (180^)/(3) = 60^ The correct option is C. The measure of the angle is 60^. 49. Step 1: Rewrite the division as a fraction. (2 - i) ÷ (1 + 2i) = (2-i)/(1+2i) Step 2: Multiply the numerator and denominator by the conjugate of the denominator. The conjugate of 1+2i is 1-2i. (2-i)/(1+2i) × (1-2i)/(1-2i) Step 3: Perform the multiplication in the numerator. (2-i)(1-2i) = 2(1) + 2(-2i) - i(1) - i(-2i) = 2 - 4i - i + 2i^2 Substitute i^2 = -1: = 2 - 5i + 2(-1) = 2 - 5i - 2 = -5i Step 4: Perform the multiplication in the denominator. This is in the form (a+bi)(a-bi) = a^2 + b^2. (1+2i)(1-2i) = 1^2 + 2^2 = 1 + 4 = 5 Step 5: Combine the simplified numerator and denominator. (-5i)/(5) = -i The correct option is A. The simplified expression is -i. 50. Step 1: Write the given equation. y = mx + c Step 2: Isolate the term containing m. Subtract c from both sides of the equation: y - c = mx Step 3: Solve for m. Divide both sides by x: (y-c)/(x) = m So, m = (y-c)/(x). The correct option is B. Making m the subject of the equation gives m = (y-c)/(x). Last free one today — make it count tomorrow, or type /upgrade for unlimited.