Question

The parallel sides of a trapezium measure 3 cm and 9 cm. the non-parallel sides measure 4 cm and 6 cm. a line parallel to the parallel sides divides the trapezium into two parts of equal perimeters. the ratio in which each of the non-parallel sides is divided is

Answer 1

Divide both sides by Take reciprocals of both sides Add 1 to both sides Replace 1 by on the left and replace 1 on the right by Combine numerators over denominators: Since AP+PD = AD = 6 and BQ+QC = BC = 4 Take reciprocals of both sides Let that ratio be = k and BQ = 4k, QC = 4 - 4k AP = 6k PD = 6 - 6k all in centimeters. Perimeter of the upper trapezoid ABQP: BQ + AB + AP + PQ Perimeter of the lower trapezoid PQCD: QC + DC + PD + PQ Equating the two perimeters: BQ + QB + AP + PQ = QC + DC + PD + PQ Subtract PQ (their common side) from both sides: BQ + QB + AP = QC + DC + PD Substitute their lengths in terms of the fraction k: 4k + 3 + 6k = (4-4k) + 9 + (6-6k) 10k + 3 = 4 - 4k + 9 + 6 - 6k 10k + 3 = 19 - 10k 20k = 16 k = = = 0.8 Therefore AP = 6k = 6(0.8) = 4.8 cm. BQ = 4k = 4(0.8) = 3.2 cm. PD = 6-6k = 6-6(0.8) = 1.2 cm. QC = 4-4k = 4-4(0.8) = 0.8 cm We want to find AP:PD = BQ:QC which is 4.8:1.2 = 48:12 = 4:1 and checking: BQ:QC = 3.2:0.8 = 32:8 = 4:1 That's the answer 4:1 Incidentally, the figures above are drawn approximately to scale. Measure them with a ruler and you will see they are fairly close to 3cm, 4cm, 9cm and 6cm. The perimeters are equal even though the upper trapezoid is obviously lots bigger in area. This demonstrates the important fact that two figures can have the same perimeter and quite different areas.