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 is the solution to your question: a) Calculate the length of the common chord AB. Step 1: Identify the given information. We have two circles with the same radius, r = 8 cm. The distance between their centers is d = 12.8 cm. The common chord AB connects the two points where the circles intersect. The line connecting the centers of the circles (let's call them C_1 and C_2) is perpendicular to the common chord AB and bisects it. Since the radii of both circles are equal, the midpoint of the chord AB (let's call it M) is also the midpoint of the line segment C_1C_2. Therefore, the distance from one center to M is h = (d)/(2) = (12.8)/(2) = 6.4 cm. Step 2: Use the Pythagorean theorem to find half the length of the chord AB. Consider the right-angled triangle C_1MA, where C_1 is the center of one circle, M is the midpoint of the chord AB, and A is one of the intersection points. C_1A is the radius, r = 8 cm (this is the hypotenuse). C_1M is the distance from the center to the chord, h = 6.4 cm. AM is half the length of the chord AB. According to the Pythagorean theorem: C_1A^2 = C_1M^2 + AM^2 8^2 = (6.4)^2 + AM^2 64 = 40.96 + AM^2 AM^2 = 64 - 40.96 AM^2 = 23.04 AM = sqrt(23.04) AM = 4.8 cm Step 3: Calculate the full length of the common chord AB. The length of the common chord AB is twice the length of AM. AB = 2 × AM AB = 2 × 4.8 cm AB = 9.6 cm b) Calculate the area of the common region to both circles. Step 4: Calculate the area of one circular segment. The common region is formed by two identical circular segments, one from each circle. The area of a circular segment is given by the formula: A_segment = r^2 ^-1((h)/(r)) - hsqrt(r^2-h^2) Where r = 8 cm and h = 6.4 cm. First, calculate the angle = ^-1((h)/(r)) in radians: = ^-1((6.4)/(8)) = ^-1(0.8) ≈ 0.6435011 radians Next, calculate sqrt(r^2-h^2): sqrt(8^2 - 6.4^2) = sqrt(64 - 40.96) = sqrt(23.04) = 4.8 cm Now substitute these values into the segment area formula: A_segment = (8^2) × (0.6435011) - (6.4) × (4.8) A_segment = 64 × 0.6435011 - 30.72 A_segment = 41.1840704 - 30.72 A_segment = 10.4640704 cm^2 Step 5: Calculate the total area of the common region. The total common area is twice the area of one circular segment. A_common = 2 × A_segment A_common = 2 × 10.4640704 cm^2 A_common = 20.9281408 cm^2 Rounding to two decimal places: A_common ≈ 20.93 cm^2 3 done, 2 left today. You're making progress.