🏗️ Part 1: Urban Planning and Construction

1. A new subdivision in Auckland is being planned. The community centre is located at point A(2, 5) and the shopping centre is at point B(8, 13). Each unit on the coordinate plane represents 100 metres.

a) Calculate the straight-line distance between the community centre and shopping centre in metres.

b) The council wants to place a bus stop exactly halfway between these two locations. Find the coordinates of the bus stop.

c) What is the actual distance in metres from the community centre to the bus stop?

🚁 Part 2: Emergency Services Navigation

2. A rescue helicopter needs to navigate between three locations during an emergency response in Wellington. The hospital is at H(1, 3), the accident site is at A(7, 11), and the police station is at P(13, 4). Each coordinate unit represents 500 metres.

a) Find the equation of the straight flight path from the hospital to the accident site.

b) The helicopter needs to fly from the accident site to the police station. Write the equation of this flight path in the form y = mx + c.

c) Calculate the total distance the helicopter travels from H → A → P in kilometres.

🏠 Part 3: Property Development

3. A property developer is designing a rectangular housing block in Christchurch. Three corners of the block have been surveyed at coordinates R(2, 1), S(10, 3), and T(8, 11).

a) To complete the rectangle, find the coordinates of the fourth corner U.

b) Verify your answer by showing that opposite sides are parallel. Calculate the gradients of all four sides.

c) Show that adjacent sides are perpendicular by demonstrating that their gradients multiply to give -1.

🛣️ Part 4: Road Engineering

4. A new highway is being constructed through Hamilton. The main road follows the line with equation 3x + 4y = 24. A service road needs to be built perpendicular to the main road, passing through the petrol station located at point G(8, 6).

a) Find the gradient of the main road.

b) Determine the gradient of the service road (perpendicular to the main road).

c) Write the equation of the service road in the form y = mx + c.

d) Find the coordinates where the service road intersects the main highway.

📡 Part 5: Telecommunications Network

5. A telecommunications company is installing fibre optic cables in Dunedin. They need to connect three cell towers located at A(1, 2), B(9, 8), and C(5, 14). Each coordinate unit represents 200 metres.

a) Calculate the total length of cable needed to connect all three towers in a triangular network (A to B to C to A) in kilometres.

b) To reduce costs, they want to install a central hub at the centroid of the triangle and connect each tower to this hub. Find the coordinates of the centroid.

c) Calculate the total cable length needed for the hub design and determine the cost savings compared to the triangular network if cable costs $150 per metre.

🎯 Part 6: Multiple Choice Applications

6. Two parallel roads in Tauranga have equations y = 2x + 3 and y = 2x - 5. What is the perpendicular distance between these roads if each unit represents 50 metres?

200√5 metres

160√5 metres

80√5 metres

400 metres

7. A straight walking track connects two lookouts at coordinates (3, 7) and (11, 1). Which of the following points lies exactly on this track?

(5, 5)

(7, 4)

(9, 2)

(6, 6)