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 calculations for the given figure. 2.1 Calculate the perimeter of rectangle ADEH. Step 1: Identify the dimensions of rectangle ADEH. The length AD = 62 mm and the width AH = 32 mm. Step 2: Use the formula for the perimeter of a rectangle. The perimeter P of a rectangle is given by P = 2 × (length + width). P_ADEH = 2 × (AD + AH) P_ADEH = 2 × (62 mm + 32 mm) P_ADEH = 2 × (94 mm) P_ADEH = 188 mm The perimeter of rectangle ADEH is 188 mm. 2.2 Calculate the perimeter of rectangle CDEF. Step 1: Identify the dimensions of rectangle CDEF. Since ADEH is a rectangle, DE = AH = 32 mm. From the figure, FE = 18 mm. Since CDEF is a rectangle, CD = FE = 18 mm. So, the length CD = 18 mm and the width DE = 32 mm. Step 2: Use the formula for the perimeter of a rectangle. P_CDEF = 2 × (CD + DE) P_CDEF = 2 × (18 mm + 32 mm) P_CDEF = 2 × (50 mm) P_CDEF = 100 mm The perimeter of rectangle CDEF is 100 mm. 2.3 Calculate the area of the shaded region within rectangle ADEH. The shaded region consists of triangle HBF and rectangle CDEF. Step 1: Calculate the length of GF. The total length HE = AD = 62 mm. We know HE = HG + GF + FE. 62 mm = 20 mm + GF + 18 mm 62 mm = 38 mm + GF GF = 62 mm - 38 mm GF = 24 mm Step 2: Calculate the area of triangle HBF. The base of the triangle is HF = HG + GF = 20 mm + 24 mm = 44 mm. The height of the triangle is BG. Since ADEH is a rectangle, BG = AH = 32 mm. The area A of a triangle is given by A = (1)/(2) × base × height. A_HBF = (1)/(2) × HF × BG A_HBF = (1)/(2) × 44 mm × 32 mm A_HBF = 22 mm × 32 mm A_HBF = 704 mm^2 Step 3: Calculate the area of rectangle CDEF. The length CD = 18 mm and the width DE = 32 mm. The area A of a rectangle is given by A = length × width. A_CDEF = CD × DE A_CDEF = 18 mm × 32 mm A_CDEF = 576 mm^2 Step 4: Calculate the total shaded area. The total shaded area is the sum of the area of triangle HBF and the area of rectangle CDEF. A_shaded = A_HBF + A_CDEF A_shaded = 704 mm^2 + 576 mm^2 A_shaded = 1280 mm^2 The area of the shaded region within rectangle ADEH is 1280 mm^2. Send me the next one 📸