The parallel sides of a trapezium and its height are in an arithmetic progression with a common difference of 4. If the height is the highest term and the area of the trapezium is 160 sq. units, find the ratio of length of greatest parallel side to that of the smallest parallel side.

Explanation

Let: Smallest parallel side = a Second parallel side = a + 4 Height = a + 8 Step 2: Set Up the Area Formula for a Trapezium The area A of a trapezium with parallel sides a and b, and height h is given by: A = (1/2) × (sum of parallel sides) × height Given: Area A = 160 Sum of parallel sides = a + (a + 4) = 2a + 4 Height = a + 8 Step 3: Substitute Values into the Area Formula 160 = (1/2) × (2a + 4) × (a + 8) Multiply both sides by 2: 320 = (2a + 4)(a + 8) Step 4: Expand and Simplify 320 = 2a2 + 20a + 32 0 = 2a2 + 20a - 288 Divide by 2: 0 = a2 + 10a - 144 Step 5: Solve the Quadratic Equation Using the quadratic formula: a = [-b ± √(b2 - 4ac)] / 2a a = [-10 ± √(102 + 4 × 144)] / 2 a = 8 (selecting the positive value as length cannot be negative) Step 6: Determine Lengths of the Parallel Sides Smallest parallel side = a = 8 Greatest parallel side = a + 4 = 12 Step 7: Calculate the Ratio Ratio = Greatest Parallel Side / Smallest Parallel Side = 12 / 8 = 3 / 2 Final Answer The ratio of the length of the greatest parallel side to that of the smallest parallel side is 3:2.