A Triangular Garden Plot with a Fence and a Path
A triangular garden plot ABC has sides AB = 13 m, BC = 14 m and AC = 15 m. The owner wants to fence the whole perimeter and then lay a straight gravel path from vertex A to the midpoint M of side BC. A landscaper claims the path AM is also the height (altitude) from A to BC and therefore splits the triangle into two equal-area halves. Fencing costs 9 leva per metre. The owner has a budget of 400 leva for fencing and the path combined, and the path costs 12 leva per metre.
1. Compute the perimeter and the fencing cost. Is the claim that AM is the altitude true? Justify using side lengths.
Read the worked answer
Perimeter = 13 + 14 + 15 = 42 m. Fencing cost = 42 × 9 = 378 leva.
Altitude test: if AM were perpendicular to BC, then by the right-triangle relation AB² − BM² would equal AC² − CM². Here BM = CM = 7 (M is the midpoint). AB² − BM² = 169 − 49 = 120. AC² − CM² = 225 − 49 = 176. These differ, so AM is not perpendicular to BC. The landscaper's claim is false: a median equals an altitude only in an isosceles triangle with the equal sides meeting at that vertex, which is not the case here.
2. The path AM does split the triangle into two equal areas even though it is not an altitude. Explain why, and compute the total cost of fencing plus the path using the length AM found from the median formula.
Read the worked answer
Triangles ABM and ACM have bases BM = CM = 7 on the same line BC, and both share the same perpendicular height from A to line BC. Equal bases and equal heights give equal areas, so AM as a median does halve the area - the landscaper's conclusion was right but the reason (altitude) was wrong.
Median length: AM² = (2AB² + 2AC² − BC²)/4 = (2×169 + 2×225 − 196)/4 = (338 + 450 − 196)/4 = 592/4 = 148. AM = √148 ≈ 12.17 m. Path cost ≈ 12.17 × 12 ≈ 146 leva. Total ≈ 378 + 146 = 524 leva, which exceeds the 400 leva budget.
3. The owner can shorten the fence by leaving one side open. Which single side should be left open to stay within the 400 leva budget while keeping the path?
Read the worked answer
Fencing two sides costs 9 leva per metre of those sides; path ≈ 146 leva.
Leave BC (14 m) open: fence 13 + 15 = 28 m, cost 252; total 252 + 146 = 398 leva - within budget. Leave AC (15 m) open: fence 13 + 14 = 27 m, cost 243; total 389 leva - within budget. Leave AB (13 m) open: fence 14 + 15 = 29 m, cost 261; total 407 leva - over budget.
So leaving AB open fails; leaving BC or AC open works, with AC open the cheapest at about 389 leva. The judgement is to test each option against the budget rather than assume the longest side is the best to drop.