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.
ScanSolve AI Answer
Here's how to solve the problem: Part (i): Show that AD = 18 cm. Step 1: Draw a perpendicular line from point D to AB. Let E be the point on AB such that DE is perpendicular to AB. This forms a rectangle EBCD and a right-angled triangle ADE. Step 2: Determine the lengths of DE and AE. Since EBCD is a rectangle, DE = BC = 14.4 cm and EB = DC = 16 cm. Then, AE = AB - EB = 26.8 - 16 = 10.8 cm. Step 3: Use the Pythagorean theorem in triangle ADE. In the right-angled triangle ADE: AD^2 = AE^2 + DE^2 AD^2 = (10.8)^2 + (14.4)^2 AD^2 = 116.64 + 207.36 AD^2 = 324 AD = sqrt(324) AD = 18 cm This shows that AD = 18 cm. Part (ii): Calculate the perimeter of the shaded region. Step 1: Identify all segments forming the perimeter. The perimeter of the shaded region consists of the lengths AB, BC, AD, and the arc length of the semi-circle. Step 2: List the known lengths. AB = 26.8 cm BC = 14.4 cm AD = 18 cm (from part i) Step 3: Calculate the arc length of the semi-circle. The diameter of the semi-circle is DC = 16 cm. The radius r = (16)/(2) = 8 cm. The arc length of a semi-circle is given by the formula r. Arc length = × 8 = 8 cm Step 4: Calculate the total perimeter. Add all the lengths together: Perimeter = AB + BC + AD + Arc length Perimeter = 26.8 + 14.4 + 18 + 8 Perimeter = 59.2 + 8 Using the value of ≈ 3.14159: Perimeter ≈ 59.2 + 8 × 3.14159 Perimeter ≈ 59.2 + 25.13272 Perimeter ≈ 84.33272 Rounding to one decimal place: Perimeter ≈ 84.3 cm The perimeter of the shaded region is 84.3 cm. Send me the next one 📸