Certifying a Triangular Community Plot
A Haryana housing board has marked out a triangular community plot on a survey sheet on which one grid unit represents 10 m. The three corners are recorded as A(2, 1), B(10, 5) and C(4, 9). The board plans a borewell point D on side BC that divides BC in the ratio BD : DC = 3 : 1, so that the water line towards the far corner C stays short. A paved path from A to D will split the plot into two maintenance zones whose annual upkeep cost of ₹56,000 is shared between residents in proportion to area. Separately, a resident has asked for a lamp post at the map point (9, 8), claimed to be 'just inside the eastern edge'. The surveyor must certify the plot area in square metres, fix the coordinates of D, and rule on the lamp post before the layout goes for approval.
1. Find the area of the plot in square metres.
Read the worked answer
Use the coordinate area formula:
Area = ½|x_A(y_B − y_C) + x_B(y_C − y_A) + x_C(y_A − y_B)| = ½|2(5 − 9) + 10(9 − 1) + 4(1 − 5)| = ½|−8 + 80 − 16| = ½(56) = 28 square units.
Scale: 1 unit = 10 m, so 1 square unit = 10 m × 10 m = 100 m².
Plot area = 28 × 100 = 2800 m².
Sanity check: the enclosing 8 × 8 unit square is 6400 m² and the plot visibly fills a little under half of it, so 2800 m² is reasonable.
2. Find the coordinates of D, and state how the ₹56,000 upkeep cost should be divided between the zones on either side of AD. Justify the division without relying only on the coordinate formula.
Read the worked answer
D divides B(10, 5) and C(4, 9) in the ratio 3 : 1:
D = ((3 × 4 + 1 × 10)/4, (3 × 9 + 1 × 5)/4) = (22/4, 32/4) = (5.5, 8).
Justification by geometry: triangles ABD and ADC share the same perpendicular height from A to the line BC, so their areas are in the ratio of their bases, BD : DC = 3 : 1.
Verification: area ABD = ½|2(5 − 8) + 10(8 − 1) + 5.5(1 − 5)| = ½|−6 + 70 − 22| = 21; area ADC = ½|2(8 − 9) + 5.5(9 − 1) + 4(1 − 8)| = ½|−2 + 44 − 28| = 7. Ratio 21 : 7 = 3 : 1 ✓ and 21 + 7 = 28 ✓.
Cost split: zone ABD pays (3/4) × 56,000 = ₹42,000; zone ADC pays (1/4) × 56,000 = ₹14,000.
3. Is the lamp post at (9, 8) inside the plot? Decide using areas, confirm with a second independent check, and name a point that does lie inside.
Read the worked answer
Area test - if the point Q(9, 8) is inside, the three sub-triangles must add up to 28 square units:
ABQ = ½|2(5 − 8) + 10(8 − 1) + 9(1 − 5)| = ½|−6 + 70 − 36| = 14 BCQ = ½|10(9 − 8) + 4(8 − 5) + 9(5 − 9)| = ½|10 + 12 − 36| = 7 CAQ = ½|4(1 − 8) + 2(8 − 9) + 9(9 − 1)| = ½|−28 − 2 + 72| = 21 Sum = 42 ≠ 28, so Q is outside the plot.
Independent check: line BC is 2x + 3y = 35 (B: 20 + 15 = 35 ✓; C: 8 + 27 = 35 ✓). For Q, 2(9) + 3(8) = 42 > 35, while for the opposite corner A, 2(2) + 3(1) = 7 < 35. Q is on the far side of BC from A, i.e. beyond the eastern edge.
A point that is inside: (6, 5), whose sub-areas are 8, 8 and 12, summing to 28 ✓. The lamp post must be relocated; (6, 5) would be certified.