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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To find the area of the compound figure and the missing angle 'x', we will break down the figure into simpler shapes and use trigonometric ratios. Part 1: Finding the Area of the Compound Figure We can decompose the figure into four basic shapes: three rectangles and one right-angled triangle. Step 1: Identify the shapes and their dimensions. Rectangle 1 (R1 - Bottom section): This rectangle spans the entire width of the base and has a height of 4 cm. Width = 4 cm + 8 cm + 6 cm = 18 cm Height = 4 cm Rectangle 2 (R2 - Left section above R1): This rectangle is on the far left, above the 4 cm height of R1. Width = 4 cm Height = 8 cm - 4 cm = 4 cm (total height on left is 8 cm, 4 cm is part of R1) Rectangle 3 (R3 - Middle section above R1): This rectangle is in the middle, above the 8 cm width segment. Width = 8 cm Height = 4 cm Triangle 1 (T1 - Right section above R1): This is a right-angled triangle on the far right, above the 6 cm width segment. Base = 6 cm Height = 4 cm Step 2: Calculate the area of each individual shape. Area of R1: A_R1 = width × height = 18 cm × 4 cm = 72 cm^2 Area of R2: A_R2 = width × height = 4 cm × 4 cm = 16 cm^2 Area of R3: A_R3 = width × height = 8 cm × 4 cm = 32 cm^2 Area of T1: A_T1 = (1)/(2) × base × height = (1)/(2) × 6 cm × 4 cm = 12 cm^2 Step 3: Calculate the total area. The total area of the compound figure is the sum of the areas of these individual shapes. A_total = A_R1 + A_R2 + A_R3 + A_T1 A_total = 72 cm^2 + 16 cm^2 + 32 cm^2 + 12 cm^2 A_total = 132 cm^2 The total area of the compound figure is 132 cm^2. Part 2: Finding the Missing Angle 'x' Angle 'x' is located in the right-angled triangle (T1) on the right side of the figure. Step 1: Identify the known sides relative to angle 'x'. The side opposite* to angle 'x' is the height of the triangle, which is 4 cm. The side adjacent* to angle 'x' is the base of the triangle, which is 6 cm. Step 2: Use the tangent trigonometric ratio. The tangent of an angle in a right-angled triangle is the ratio of the length of the opposite side to the length of the adjacent side. (x) = oppositeadjacent (x) = 4 cm6 cm (x) = (2)/(3) Step 3: Calculate the angle 'x' using the inverse tangent function. x = ((2)/(3)) x ≈ 33.690067^ Step 4: Round the angle to one decimal place. x ≈ 33.7^ The missing angle 'x' is 33.7^.