I teach International Baccalaureate Mathematics (and I actually spelled baccalaureate correctly without spell-check …). Their curriculum contains a topic that I had not even explored, Voronoi diagrams. It turns out these are a great way to practice and apply concepts and procedures in both Algebra and Geometry. Of course, there are plenty of non-academic applications, but I’m a teacher, so I care about having engaging challenges for students.
If you’re already familiar with the basics, skip ahead to the problem. If not, here are the two key facts.
Voronoi diagrams are a tesselation of 2D space. Each area is called a cell. Each cell has a special point within it such that every other location within a cell is closer to the special point than a neighboring cell’s special point. These special points are called cell points, generator points, and site points, depending on the application.
In the image above, point M is closer to C than it is to A or B. This is true because m resides in Cell C.The boundary lines separating two cells are perpendicular bisectors of the segment connecting adjacent cell points.
If you’d like to learn more about Voronoi Diagrams, here’s a short video.
Here is this week’s question.
Clues
Clue #1: The boundary line separating adjacent cells is the perpendicular bisector to the segment connecting adjacent cells’ cell points. In our case, that would be connecting (4, 8) to our unknown coordinate.
Clue #2: The intersection of the boundary line and the segment connecting (4, 8) to the unknown point is the midpoint of the ends of the segment (4, 8) and (?, ?). To find the midpoint you can set the lines equal and solve for x, then find y, or you can use a graphing calculator.
Clue #3: Use the midpoint formula to find the missing values of x and y.
Solution
Video
Equation of the Segment
We know that our unknown point and (4, 8) are on a segment that is perpendicular to, and bisected by, the boundary line given. So, to begin we will discover the line of the segment connecting (4, 8) and the unknown point. Let’s begin with the slope, which will be 3 as that is the negative reciprocal of the slope of the boundary line.
Knowing the slope is 3, and it passes through the point (4, 8), let’s plug those values into y = mx + b, to find the equation of the line.
The equation of the segment connecting (4, 8) and the unknown coordinate is y = 3x – 4. Nice!
The Midpoint
Where the boundary and our segment intersect is the midpoint of the segment. That will allow us to find the missing coordinate. To find the midpoint we will set the lines equal and solve for x. This is the x – coordinate of the midpoint. We can plug this value into either equation to find the y – coordinate.
We have found that the midpoint is at (3, 5).
Using the MidPoint Formula to Find (?, ?)
We will write the midpoint formula, plug in the values (4, 8) for x1 and y1, and then (3, 5) as the midpoint, and then solve for the unknown values.
Resources and Reflections
First off, this is a great topic to teach students that are past Algebra 1 level, but need some work on linear equations (so pretty much all students past Algebra 1). It is a fresh way to practice slope, writing equations, solving linear equations, using formulas, and all of that good old linear equations stuff.
If you would like to use this problem as either a lesson or a challenge problem, you can grab a copy of the PowerPoint, student handout, and Google Forms quiz containing and supporting this very problem. Click here to download your copy.