Splitting a Triangular Field with a Bund Parallel to One Side
A farmer in Ganjam owns a triangular field ABC in which AB = AC = 50 m and BC = 60 m. To grow two crops he wants to divide the field by a straight earthen bund DE parallel to BC, with D on AB and E on AC, so that the small triangle ADE near the apex A carries vegetables and the trapezium DBCE carries paddy. The village accountant has recorded the area of the whole field as 1,200 m². The farmer first asks where D must be placed so that triangle ADE has an area of 300 m². A neighbour then advises him to put D 30 m from A, arguing that since 30 m is 60% of AB, the small triangle will take 60% of the field. Finally, the farmer already owns fencing panels totalling 24 m in length and wants to know what area triangle ADE will have if the bund is exactly 24 m long.
1. Find the position of D so that triangle ADE has area 300 m², and state the length of the bund DE in that case.
Read the worked answer
Since DE ∥ BC, ∠ADE = ∠ABC and ∠AED = ∠ACB (corresponding angles), so △ADE ~ △ABC by AA similarity.
For similar triangles, ratio of areas = (ratio of corresponding sides)²:
Area(ADE)/Area(ABC) = (AD/AB)²
300/1200 = 1/4 = (AD/50)² → AD/50 = 1/2 (positive root) → AD = 25 m.
So D is the midpoint of AB. Then DE/BC = AD/AB = 1/2 → DE = 30 m, and E is likewise the midpoint of AC.
Check with the field's actual dimensions: height from A to BC = √(50² − 30²) = 40 m, so Area(ABC) = ½ × 60 × 40 = 1200 m² ✓, and the small triangle has area ½ × 30 × 20 = 300 m² ✓.
2. Evaluate the neighbour's advice: if AD = 30 m, does triangle ADE really take 60% of the field?
Read the worked answer
AD/AB = 30/50 = 3/5, so the linear scale factor is 3/5.
Area ratio = (3/5)² = 9/25 = 0.36, i.e. 36%, not 60%.
Area(ADE) = 0.36 × 1200 = 432 m², and the paddy trapezium gets 1200 − 432 = 768 m².
Why the neighbour is wrong: lengths scale by k but areas scale by k². Taking 60% of the side does not take 60% of the area; it takes only 36%. The advice would leave the farmer with a vegetable plot far smaller than he planned and a paddy area larger than intended.
(Cross-check with the 24 m / 25 m results: at AD = 25 m the area is 300 m²; at AD = 30 m it is 432 m² - both consistent with the square law.)
3. If the bund DE must be exactly 24 m long, find the area of triangle ADE and the length AD.
Read the worked answer
DE corresponds to BC, so the scale factor is DE/BC = 24/60 = 2/5.
Area(ADE) = (2/5)² × Area(ABC) = (4/25) × 1200 = 192 m².
AD = (2/5) × AB = (2/5) × 50 = 20 m, and similarly AE = 20 m.
Check: height of △ADE = (2/5) × 40 = 16 m, so area = ½ × 24 × 16 = 192 m² ✓.
Decision for the farmer: a 24 m bund gives only 192 m² of vegetables, i.e. 16% of the field, so if he wants the vegetable plot at roughly a quarter of the field he should not be constrained by the 24 m of panels he owns - he needs a 30 m bund.