How To Draw Straight Lines Without A Ruler

Geometry without the tools

One of the best qualities someone can take is resourcefulness. Fifty-fifty when a trouble seems insurmountable with the electric current gear up of tools, resourceful people can observe a solution effectually it. The path to solve the problem might not be as directly and anticipated as it would have been with the correct knowledge, instruments, emotional weather. Notwithstanding, 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 all the same 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 usa to be resourceful and find a style effectually our forgetfulness, those days we might have learnt more than than but the Pythagorean Theorem.

Today we will focus on two tasks that would be trivial with the correct set of tools, but that end upwards existence 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 nonetheless working. Help him construct an equilateral triangle with side length 10cm.

In other words, nosotros accept to depict an equilateral triangle of a 10 cm size simply we tin can only describe circles of 4 cm radius.

If such constraints were not in identify, nosotros would describe a 10 cm line from point A to point B. Then, using the compass open to ten cm, we would describe two circles centered in A and in B. One of their intersections would be the last vertex of our triangle. Connecting the three points with our straight edge would give usa the triangle.

If our toolbox were consummate, nosotros would draw an equilateral triangle in this way.

But our toolbox is not consummate. We need to find a manner to tackle the problem without drawing circles of 10 cm radius. What can we still make up one's mind given our constraints? Give information technology a thought earlier 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 volition be the beginning side of the triangle. We tin also describe lines incident at 60' on our main segment, since we tin draw smaller equilateral triangles of 4 cm side.

Check out the post-obit video to see how we solved the problem:

Our arroyo to solving the rusty compass problem

This is just i possible manner of solving the problem. Can you lot detect a amend style to solve it? Recall near it, only for now we will move to the next problem.

The Cleaved Ruler

The ready of the second problem is as follows:

Yumi'southward ruler broke into fiddling pieces, so she can merely draw lines iii cm long. Fortunately, her compass is still working. She has 2 points on her paper approximately 10 cm autonomously. Help her construct the straight line joining those 2 points

In other words, nosotros want to draw a 10 cm segment simply we only take a 3 cm straight edge. This fourth dimension we do have a working compass, different in the previous example. Our starting state of affairs includes ii points, A and B, about ten cm apart. We need to discover more points on the line betwixt them in such a style that nosotros can connect them using our small ruler.

Many approaches are possible, then call back most how you would notice boosted points on the line between A and B. New points can be defined past the intersections of circles. Once you lot detect a feasible strategy, attempt to perfect information technology and become rid of unnecessary steps.

Hint: Our arroyo relies on the fact that the intersections between circles centered at 2 stock-still points, but with variable radius, prevarication on the same line.

Check out the following video to see how we solved the trouble:

Can you find a better style to solve information technology?

Figuring out what we can withal do given the current limitations is the showtime of the beginning role to the solution. The 2d part consists on getting rid of all the unnecessary steps, the redundancies. Nosotros want our final path to be as smooth, as direct and as elegant equally possible.

In this case, we had to exist resourceful to solve a trouble that would rarely happen in real life. Still, nosotros got some insight on the difficulty of approaching a solution when thinking inside the box is non immune. This blazon of insight tin help us navigate situations where beingness resourceful is not a matter of selection.