AI HL · 12.8
0 of 19 skills done
Sites, cells, edges and vertices
12.8.3
Screenshot and video to come
Screenshot and video to come
Drag the sites A and B. The working finds the perpendicular bisector line by line. Check that it matches the line on the grid.
Drag A or B.
Drag the sites, add one or take one away, and watch the cells change. Drag the orange point P to see which site is nearest to it.
Drag a site or the point P.
The diagram shows the Voronoi diagram for the sites \(A(3, 2)\), \(B(11, 6)\), \(C(5, 8)\). Name the parts, and write down the number of edges.
Have a go first. Then open the working one line at a time.
A, B and C
The sites are the given points, marked with dots.
3 cells
Each site has one cell: all the points closer to it than to the other two.
3 edges
There is one edge between each pair of cells. Each lies on the perpendicular bisector of two sites.
1 vertex at \((7, 4)\)
The three edges meet here. It is \(4.47\) from each site.
Draw the Voronoi diagram for the sites \(A(8, 3)\), \(B(12, 5)\), \(C(6, 11)\), \(D(2, 5)\), \(E(6, 7)\). Find the y-coordinate of the vertex where the cells of B, C and E meet.
Have a go first. Then open the working one line at a time.
Neighbouring pairs: AB \((10, 4)\), AD \((5, 4)\), AE \((7, 5)\), BC \((9, 8)\), BE \((9, 6)\), CD \((4, 8)\), CE \((6, 9)\), DE \((4, 6)\)
Mark each midpoint on the grid. Neighbours are sites with no other site in between.
AB: the sites have gradient \(\frac{1}{2}\), so the edge has gradient \(-2\); AD: the sites have gradient \(-\frac{1}{3}\), so the edge has gradient \(3\); AE: the sites have gradient \(-2\), so the edge has gradient \(\frac{1}{2}\) ...
Through each midpoint draw a line at a right angle to the line joining the two sites.
From the calculator: the edges of BC and BE meet at \((10, 9)\)
Solve the equations of two edges together. Stop the edges at the vertices: \((\frac{5}{2}, 9)\), \((5, 4)\), \((9, 6)\), \((10, 9)\).
All five cells drawn
Check: \((10, 9)\) is \(4.47\) from B, C and E.
The sites are \(A(0, 5)\), \(B(3, 2)\), \(C(4, 10)\), \(D(10, 2)\), \(E(4, 5)\). The dashed lines are three edges of the cell containing E. (a) Find the equation of the line that forms the final edge, in the form \(px+qy=r\). (b) Sketch the cell containing E. (c) What can you say about the point \((5, 4)\), which lies in the cell?
Have a go first. Then open the working one line at a time.
The edge between D and E
The dashed lines are the edges of E with A, B and C. The cell of E also borders D.
Midpoint: \((7, \frac{7}{2})\); gradient of DE \(-\frac{1}{2}\); edge gradient \(2\); \(4x-2y=21\)
The edge is the perpendicular bisector of D and E. Multiply to clear fractions.
Vertices \((9, \frac{15}{2})\), \((2, \frac{15}{2})\), \((2, 4)\), \((\frac{13}{2}, \frac{5}{2})\)
The final edge closes the cell into a quadrilateral. Shade it.
\(QE=\sqrt{2}=1.41\); next nearest \(QB=\sqrt{8}=2.83\)
A point in the cell of E is closer to E than to any other site.
The sites are \(A(3, 5)\), \(B(7, 5)\), \(C(5, 1)\), \(D(5, 9)\). Do the cells of A and B share an edge? Do the cells of C and D?
Have a go first. Then open the working one line at a time.
M = \((5, 5)\); \(AM=2\), \(BM=2\), \(CM=4\), \(DM=4\)
A and B are the closest sites to M. No other site gets in between.
They share an edge
The edge lies on the line x = 5, around the midpoint.
The midpoint is also \((5, 5)\)
A and B are both closer to this point than C and D are.
They do not share an edge
The cells of A and B come between the cells of C and D.
Give the coordinates of four sites for each case: (a) a diagram with two vertices, each with three edges; (b) a diagram with one vertex with four edges; (c) a diagram where one site is completely surrounded by the cells of the other three.
Have a go first. Then open the working one line at a time.
Sites \(A(1, 7)\), \(B(5, 7)\), \(C(5, 2)\), \(D(8, 4)\); vertices \((3, \frac{9}{2})\) and \((\frac{11}{2}, \frac{9}{2})\)
Four sites that do not lie on one circle give two vertices.
Sites \(A(2, 1)\), \(B(8, 1)\), \(C(8, 5)\), \(D(2, 5)\); vertex \((5, 3)\)
The four corners of a rectangle are all \(3.61\) from the centre, so all four cells meet there.
Sites \(A(9, 10)\), \(B(6, 3)\), \(C(2, 11)\), \(D(6, 7)\); D inside triangle ABC
D has an edge with each of A, B and C, so its cell is a triangle: vertices \((11, 5)\), \((\frac{11}{2}, \frac{21}{2})\), \((0, 5)\).
A nature reserve is a rectangle 12 km by 8 km. Four ranger stations are at \(A(5, 5)\), \(B(2, 3)\), \(C(6, 1)\) and \(E(5, 3)\), where one unit is 1 km. The cell of E is a triangle with vertices \((3.5, 1)\), \((3.5, 4)\) and \((9.5, 4)\). Find the area of the cell of E, and the percentage of the reserve that is nearer to E than to any other station.
Have a go first. Then open the working one line at a time.
Base along \(y=4\): \(9.5-3.5=6\). Height from \((3.5, 1)\) up to \(y=4\): \(3\)
Two of the vertices share the line y = 4, so take that side as the base. The right angle is at (3.5, 4), so the height is the vertical side.
\(\frac{1}{2}\times6\times3=9\) km²
Area of a triangle is half the base times the height.
Reserve: \(12\times8=96\) km². \(\frac{9}{96}\times100=9.375\), so \(9.38\%\)
Divide the cell by the whole region. Give 3 significant figures.