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 answers to your math questions. 1) Geometry Problem Given: Two circles with equal radii, r = 9 cm. Centers A and B. Angle CAD = 80^. i) Calculate the area of Sector CAD. Step 1: Identify the formula for the area of a sector. The area of a sector is given by the formula: A_sector = ()/(360^) r^2 where is the central angle in degrees and r is the radius. Step 2: Substitute the given values into the formula. Given = 80^ and r = 9 cm. A_sector = (80^)/(360^) × × (9 cm)^2 Step 3: Simplify the expression. A_sector = (2)/(9) × × 81 cm^2 A_sector = 2 × 9 × cm^2 A_sector = 18 cm^2 Using ≈ 3.14159: A_sector ≈ 18 × 3.14159 cm^2 A_sector ≈ 56.55 cm^2 The area of Sector CAD is 18 cm^2 ≈ 56.55 cm^2. ii) Calculate the area of Triangle CAD. Step 1: Identify the formula for the area of a triangle given two sides and the included angle. Triangle CAD is an isosceles triangle with sides AC = AD = r = 9 cm and the included angle CAD = 80^. The area of a triangle is given by the formula: A_triangle = (1)/(2)ab C where a and b are the lengths of two sides and C is the included angle. Step 2: Substitute the given values into the formula. Given a = 9 cm, b = 9 cm, and C = 80^. A_triangle = (1)/(2) × 9 cm × 9 cm × (80^) Step 3: Simplify the expression. A_triangle = (81)/(2) (80^) cm^2 A_triangle = 40.5 (80^) cm^2 Using (80^) ≈ 0.9848: A_triangle ≈ 40.5 × 0.9848 cm^2 A_triangle ≈ 39.88 cm^2 The area of Triangle CAD is 40.5 (80^) cm^2 ≈ 39.88 cm^2. iii) Calculate the area of the shaded region. Step 1: Understand the shaded region. The shaded region is the area of intersection of the two circles (the lens shape). This area is composed of two identical segments, one from each circle. The area of one segment (e.g., from circle A, formed by chord CD) is the area of Sector CAD minus the area of Triangle CAD. Step 2: Calculate the area of one segment. A_segment = A_sector - A_triangle A_segment = 18 cm^2 - 40.5 (80^) cm^2 A_segment ≈ 56.55 cm^2 - 39.88 cm^2 A_segment ≈ 16.67 cm^2 Step 3: Calculate the total shaded area. Since the two circles are identical and the intersection is symmetrical, the total shaded area is twice the area of one segment. A_shaded = 2 × A_segment A_shaded = 2 × (18 - 40.5 (80^)) cm^2 A_shaded = (36 - 81 (80^)) cm^2 Using the approximate values: A_shaded ≈ 2 × 16.67 cm^2 A_shaded ≈ 33.34 cm^2 The area of the shaded region is (36 - 81 (80^)) cm^2 ≈ 33.34 cm^2. 2) Second Problem The second problem is incomplete in the image provided. Please send the full question and diagram for "2) The figure shows a circle QOT is the diameter and SRT = 37^". 3 done, 2 left today. You're making progress.