Checking a Contractor's Area Claim on a Mapped Playground
A surveyor maps the four corners of a school playground on graph paper, using the school gate as origin and taking one grid unit as 5 metres. Going round the boundary he records A(1, 2), B(7, 2), C(9, 6) and D(3, 6). The plot is to be levelled and sown with grass. A contractor glances at the sketch, calls the plot 'practically a rectangle of about 750 square metres' and quotes ₹9,000 at the rate of ₹12 per square metre. Before signing the cheque, the headmaster asks the mathematics teacher to check the figure. The peon's note in the file claims that the diagonals of the plot bisect each other at the point (5, 4), but nobody has verified it. The teacher must also tell the contractor what the shape really is, because the levelling machine lays straight parallel strips, which suits a parallelogram but not a general quadrilateral.
1. Classify quadrilateral ABCD using side lengths, diagonals and their mid-points. Is it a rectangle, a parallelogram, a rhombus, or none of these?
Read the worked answer
Compute the sides with the distance formula:
AB = √((7−1)² + (2−2)²) = √36 = 6 DC = √((9−3)² + (6−6)²) = √36 = 6 AD = √((3−1)² + (6−2)²) = √(4+16) = √20 ≈ 4.47 BC = √((9−7)² + (6−2)²) = √(4+16) = √20 ≈ 4.47
AB and DC are both horizontal and equal in length, so one pair of opposite sides is equal and parallel → ABCD is a parallelogram. AD = BC confirms the second pair.
Diagonals: AC = √((9−1)² + (6−2)²) = √(64+16) = √80 ≈ 8.94; BD = √((3−7)² + (6−2)²) = √(16+16) = √32 ≈ 5.66. The diagonals are unequal, so it is not a rectangle.
Adjacent sides 6 and 4.47 are unequal, so it is not a rhombus.
Mid-point check: mid-point of AC = ((1+9)/2, (2+6)/2) = (5, 4); mid-point of BD = ((7+3)/2, (2+6)/2) = (5, 4). They coincide, so the diagonals bisect each other - the standard parallelogram test. The peon's note is correct.
Conclusion: a parallelogram (not a rectangle), so the machine's parallel strips are suitable.
2. Find the true area of the playground in square metres.
Read the worked answer
Take AB as the base. Its length is 7 − 1 = 6 units.
The height of a parallelogram is the perpendicular distance between the two parallel sides. AB lies on the line y = 2 and DC lies on the line y = 6, so the perpendicular distance is simply 6 − 2 = 4 units. (Do not use the slant side AD ≈ 4.47 here.)
Area = base × perpendicular height = 6 × 4 = 24 square units.
Now convert the scale. 1 unit = 5 m, so 1 square unit = 5 × 5 = 25 m² (not 5 m²).
True area = 24 × 25 = 600 m².
3. Compute the fair cost of grass and evaluate the contractor's ₹9,000 quote. Where is the ₹750 m² figure likely to have come from?
Read the worked answer
Fair cost = area × rate = 600 × 12 = ₹7,200.
The contractor's ₹9,000 is ₹9,000 − ₹7,200 = ₹1,800 too high, an overcharge of 1,800/7,200 = 25%.
Where 750 m² comes from: 750 m² corresponds to 750 ÷ 25 = 30 square units, whereas the true area is 6 × 4 = 24 square units. Thirty square units is 6 × 5 - that is, the height has been taken as 5 units instead of the perpendicular height of 4, most likely by pacing or scaling off the slanted edge AD, or by counting grid lines rather than grid spaces across the top of the plot.
Advice to the headmaster: pay on the measured area, ₹7,200 at ₹12 per m², and ask the contractor to re-measure with the perpendicular height between the parallel sides before signing anything.