One of the best qualities someone can have is resourcefulness. Even when a problem seems insurmountable with the current set of tools, resourceful people can find a solution around it. The path to solve the problem might not be as direct and predictable as it would have been with the right knowledge, instruments, emotional conditions. Still, the absence of the perfect tools should rarely stop you.
Geometry is one of those fields that relies heavily on a well-defined set of tools. I still remember how my teacher would threaten to ‘write a note to our parents’ if we forgot the compass or the appropriate ruler. If instead she had told us to be resourceful and find a way around our forgetfulness, those days we might have learnt more than just the Pythagorean Theorem.
Today we will focus on two tasks that would be trivial with the right set of tools, but that end up being more interesting if we slightly modify our toolbox.
The Rusty Compass
The set up of our first problem is as follows:
Nyoman’s compass is rusted into a fixed position, so it can only draw circles of radius 4cm. Fortunately, his ruler is still working. Help him construct an equilateral triangle with side length 10cm.
In other words, we have to draw an equilateral triangle of a 10 cm size but we can only draw circles of 4 cm radius.
If such constraints were not in place, we would draw a 10 cm line from point A to point B. Then, using the compass open to 10 cm, we would draw two circles centered in A and in B. One of their intersections would be the final vertex of our triangle. Connecting the three points with our straight edge would give us the triangle.
Press enter or click to view image in full size
But our toolbox is not complete. We need to find a way to tackle the problem without drawing circles of 10 cm radius. What can we still determine given our constraints? Give it a thought before reading further.
Hint: The approach we will follow is based on the fact that we can determine a segment length of 10 cm in a straight line, given a compass with 4 cm radius. This segment will be the first side of the triangle. We can also draw lines incident at 60' on our main segment, since we can draw smaller equilateral triangles of 4 cm side.
Check out the following video to see how we solved the problem:
This is only one possible way of solving the problem. Can you find a better way to solve it? Think about it, but for now we will move to the next problem.
The Broken Ruler
The set up of the second problem is as follows:
Yumi’s ruler broke into little pieces, so she can only draw lines 3 cm long. Fortunately, her compass is still working. She has two points on her paper approximately 10 cm apart. Help her construct the straight line joining those two points
In other words, we want to draw a 10 cm segment but we only have a 3 cm straight edge. This time we do have a working compass, unlike in the previous case. Our starting situation includes two points, A and B, about 10 cm apart. We need to find more points on the line between them in such a way that we can connect them using our small ruler.
Many approaches are possible, so think about how you would find additional points on the line between A and B. New points can be defined by the intersections of circles. Once you find a feasible strategy, try to perfect it and get rid of unnecessary steps.
Hint: Our approach relies on the fact that the intersections between circles centered at two fixed points, but with variable radius, lie on the same line.
Check out the following video to see how we solved the problem:
Can you find a better way to solve it?
Figuring out what we can still do given the current limitations is the start of the first part to the solution. The second part consists on getting rid of all the unnecessary steps, the redundancies. We want our final path to be as smooth, as direct and as elegant as possible.
In this case, we had to be resourceful to solve a problem that would rarely happen in real life. Still, we got some insight on the difficulty of approaching a solution when thinking inside the box is not allowed. This type of insight can help us navigate situations where being resourceful is not a matter of choice.