U3 Question 3 AA SL Paper 2
Question 1
The Voronoi diagram shows four supermarkets represented by points with coordinates A(0, 0), B(6, 0), C(0, 6), and D(2, 2). The vertices X, Y, Z are also shown. All distances are measured in kilometers. Below are the steps to solve the problem:
a) Find the midpoint of [BD]
The coordinates of B are (6, 0) and the coordinates of D are (2, 2). The formula for the midpoint of a line segment is:
Midpoint = ((x₁ + x₂)/2 , (y₁ + y₂)/2)
For [BD]:
Midpoint = ((6 + 2)/2 , (0 + 2)/2)
Midpoint = (4, 1)
b) Find the equation of (XZ), the perpendicular bisector of DB.
The slope of DB is found using the formula slope = (y₂ - y₁) / (x₂ - x₁). For D(2, 2) and B(6, 0):
Slope of DB = (0 - 2) / (6 - 2) = -2 / 4 = -1/2
The perpendicular bisector will have a slope that is the negative reciprocal. Thus, the slope of XZ is 2. The line passes through the midpoint of [BD], which is (4, 1). Using the point-slope form of a line equation y - y₁ = m(x - x₁):
y - 1 = 2(x - 4)
Equation of XZ: y = 2x – 7
c) The equation of (XY) is y = 2 – x and the equation of (YZ) is y = 0.5x + 3.5.
Find the coordinates of X
This can be done using GDC or an algebraic mathod.
The equation of (XY) is y = 2 - x and the equation of (XZ) is y = 2x - 7. To find their intersection (X), solve the system of equations:
2 - x = 2x - 7
2 + 7 = 2x + x
9 = 3x
x = 3
Substitute x = 3 into y = 2 - x:
y = 2 - 3
y = -1
Coordinates of X = (3, -1)
d) The coordinates of Y are (-1, 3) and the coordinates of Z are (7 , 7).
Determine the exact length of [YZ]
The coordinates of Y are (-1, 3) and Z are (7, 7). The distance between two points is given by:
Distance = √((x₂ - x₁)² + (y₂ - y₁)²)
For Y(-1, 3) and Z(7, 7):
Distance = √((7 - (-1))² + (7 - 3)²)
Distance = √((7 + 1)² + (4)²)
Distance = √(8² + 4²)
Distance = √(64 + 16)
Distance = √80
e) Given that the exact length of [XY] is √32 . Find the size of XŶZ in degrees.
From the problem, we know:
- Length of [XY] = √32
- Length of [YZ] = √80
- Length of [XZ] is calculated as follows:
For X(3, -1) and Z(7, 7):
Distance = √((7 - 3)² + (7 - (-1))²)
Distance = √((4)² + (8)²)
Distance = √(16 + 64)
Distance = √80
Using the cosine rule:
cos(∠XŶZ) = (XY² + XZ² - YZ²) / (2 × XY × XZ)
Substitute the values:
cos(∠XŶZ) = (32 + 80 - 80) / (2 × √32 × √80)
cos(∠XŶZ) = 32 / (2 × √32 × √80)
cos(∠XŶZ) = 32 / (16 √10)
cos(∠XŶZ) = 2 / √10
∠XŶZ = cos⁻¹(2 / √10)
∠XŶZ ≈ 71.6°
f) Find the area of triangle XYZ
The area of a triangle given vertices is:
Area = 0.5 × XY × YZ × sin(∠XŶZ)
Substitute the values:
Area = 0.5 × √32 × √80 × sin(71.6°)
Using sin(71.6°) ≈ 0.945:
Area ≈ 0.5 × √32 × √80 × 0.945
Area ≈ 0.5 × 16 × 0.945
Area ≈ 7.56 km²
g)A town planner believes that the larger the area of the Voronoi cell XYZ, the more people will shop at supermarket D.
State one criticism of this interpretation.
The town planner assumes that a larger Voronoi cell directly correlates to a higher number of shoppers. However, the size of the Voronoi cell does not account for population density, access to transportation, or other factors that influence where people choose to shop.