Four Plots and a Diagonal Pipeline
A land surveyor records four corner stakes of a quadrilateral plot in metres relative to a gate at the origin: A(1, 2), B(7, 4), C(9, 10) and D(3, 8), taken in order around the boundary. The owner claims the plot is a parallelogram and wants a straight pipeline along diagonal AC, with a second pipeline along BD crossing it at a central junction. A contractor says that because AB and CD are equal in length, the plot must be a rectangle; another says equal opposite sides alone do not guarantee a rectangle. The surveyor must verify the shape using displacement vectors, calculate the area for costing at Rs 1,200 per square metre, and find the crossing point of the diagonals for the junction.
1. Compute vectors AB, DC, AD and BC. Show whether ABCD is a parallelogram, and test whether it is a rectangle.
Read the worked answer
Using coordinates:
AB = (7 − 1, 4 − 2) = (6, 2). DC = C − D = (9 − 3, 10 − 8) = (6, 2). AD = (3 − 1, 8 − 2) = (2, 6). BC = (9 − 7, 10 − 4) = (2, 6).
Since AB = DC, one pair of opposite sides is equal and parallel. Since AD = BC, the other pair is also equal and parallel. Therefore ABCD is a parallelogram.
To test for a rectangle, check whether adjacent sides are perpendicular. AB · AD = 6(2) + 2(6) = 12 + 12 = 24, which is not zero. So the adjacent sides are not perpendicular, and ABCD is not a rectangle. The contractor’s inference from equal opposite sides is invalid: equal opposite sides make a parallelogram, not necessarily a rectangle.
2. Find the area of ABCD and the cost of the plot at Rs 1,200 per square metre.
Read the worked answer
For vectors AB = (6, 2) and AD = (2, 6), the area of the parallelogram is the magnitude of the two-dimensional cross product:
Area = |AB × AD| = |6(6) − 2(2)| = |36 − 4| = 32 m².
The cost at Rs 1,200 per square metre is 32 × 1,200 = Rs 38,400.
The calculation is consistent with the shape: the plot is a parallelogram but not a rectangle, so using length × breadth from a rectangle assumption would be incorrect.
3. Find the intersection of diagonals AC and BD. What geometric property does your result confirm, and what does it mean for the pipeline junction?
Read the worked answer
The midpoint of AC is ((1 + 9)/2, (2 + 10)/2) = (5, 6). The midpoint of BD is ((7 + 3)/2, (4 + 8)/2) = (5, 6).
Both diagonals have the same midpoint, so they bisect each other at (5, 6). This confirms the parallelogram property that the diagonals bisect each other. For the pipelines, the junction should be placed at (5, 6); it is the midpoint of AC and also the midpoint of BD, so the two diagonal pipelines meet exactly at the centre of the plot.