Triangular Plaza on a Grid
Three proposed community facilities are marked on a grid where one unit represents 10 m: a playground at A(−2, 1), a sports hall at B(4, 5) and a clinic at C(8, −1). The council wants to link them with straight paths and turn the triangle ABC into a paved plaza. A landscape architect claims the triangle is a right-angled isosceles triangle, which would allow a simple square design with a fourth corner D. The paving contractor charges €18 per square metre plus 13.5% VAT. Before approving the design, the council needs the area of the triangle in square metres and the cost including VAT. A councillor also asks whether the midpoint of AC is the same distance from A, B and C, because a central information point is planned there.
1. Show that triangle ABC is right-angled and isosceles, and find the fourth vertex D that would complete the square ABCD.
Read the worked answer
Slope of AB = (5 − 1)/(4 − (−2)) = 4/6 = 2/3. Slope of BC = (−1 − 5)/(8 − 4) = −6/4 = −3/2. The product is (2/3)(−3/2) = −1, so AB is perpendicular to BC and the right angle is at B. Length AB = √(6² + 4²) = √52. Length BC = √(4² + (−6)²) = √52. The two legs are equal, so the triangle is right-angled and isosceles. For square ABCD in order, D = A + C − B = (−2, 1) + (8, −1) − (4, 5) = (2, −5). Check: AD = (4, −6), which equals BC, and DC = (6, 4), which equals AB, so the square conditions are satisfied.
2. Find the area of the triangle in square metres and the total cost of paving including 13.5% VAT.
Read the worked answer
Using the right angle at B, area = ½ × AB × BC = ½ × √52 × √52 = ½ × 52 = 26 square grid units. Each unit represents 10 m, so one square unit represents 10 × 10 = 100 m². The area is 26 × 100 = 2600 m². Cost before VAT = 2600 × €18 = €46,800. VAT = 13.5% of €46,800 = 0.135 × 46,800 = €6,318. Total cost = €46,800 + €6,318 = €53,118.
3. Find the midpoint of AC and verify whether it is the same distance from A, B and C. Explain why this result is expected.
Read the worked answer
Midpoint M of AC = ((−2 + 8)/2, (1 + (−1))/2) = (3, 0). Distances in grid units: MA = √((−2 − 3)² + (1 − 0)²) = √(25 + 1) = √26. MB = √((4 − 3)² + (5 − 0)²) = √(1 + 25) = √26. MC = √((8 − 3)² + (−1 − 0)²) = √(25 + 1) = √26. All three are equal, so M is equidistant from A, B and C. In metres, each distance is 10√26 ≈ 51.0 m. This is expected because ABC is right-angled at B, and the midpoint of the hypotenuse of a right triangle is the circumcentre, so it is equidistant from all three vertices.