Shortcut Trick In a trapezium, the ratio of the lengths of the parallel sides is equal to the ratio in which the diagonals divide each other. Ratio of parallel sides = Ratio of diagonal segments = 2 : 7 Given larger parallel side (7 parts) = 42 cm 1 part = 42 ÷ 7 = 6 cm Smaller parallel side (2 parts) = 2 × 6 = 12 cm ∴ The correct answer is 12 cm. Alternate Method Given: Ratio of segments of diagonals = 2 : 7 Length of larger parallel side = 42 cm Formula Used: In a trapezium ABCD with AB || CD and diagonals intersecting at O, ΔAOB ∼ ΔCOD (by AAA similarity). Therefore, AB/CD = AO/CO = BO/DO. Calculations: ⇒ Let the parallel sides be a and b, where b = 42 cm. ⇒ According to the similarity property: a / b = 2 / 7 ⇒ a / 42 = 2 / 7 ⇒ a = (2 × 42) ÷ 7 ⇒ a = 2 × 6 ⇒ a = 12 cm ∴ The correct answer is 12 cm. Related Concepts: Area of Trapezium The area of a trapezium is calculated as 1/2 × (sum of parallel sides) × height, or Area = 1/2 × (a + b) × h. Median of a Trapezium The line segment joining the midpoints of the non-parallel sides is called the median, and its length is the average of the parallel sides: Median = (a + b) ÷ 2. Similarity Property The triangles formed by the intersection of diagonals and the parallel sides (top and bottom triangles) are always similar by AAA criteria because of alternate interior angles.