Thread: Arc length and chord length

Does anyone know how to find the length of a chord knowing the radius of the circle and the arc length? Without using trigonometry.

-Dan

Do you have a specific problem?

All right, why don't I just give you the whole thing.

30 knights sit at King Arthur's round table. The table has a radius of 12 ft. Each knight has a lance that is 10 ft long. As King Arthur walks by each knight raises his lance and taps the lance of those it can reach. How many neighboring lances can one knight touch?

The student that asked this problem (elsewhere) is in 9th grade doing Geometry, and thus cannot use Trigonometry. I'm stumped!

-Dan

I suspect this might be considered cheating, but drawing a reasonably accurate scale diagram and using a piece of string to measure the arc length could work.

I think the only way to solve the problem without using trigonometry is to draw a diagram. What you do is draw a circle of radius 12, mark 30 equally spaced points on the circumference, then draw a circle of radius 20 with any one of these points as centre. (Each knight can touch the lance of another knight up to 20 ft away if both knights use their lances.) Then you just count how many points are contained in the second circle. (I think the answer is 18.)

It seems to me that in order to be able to find the length that number (the length) needs to be a constructible number if you are only using geometry without triginometry. But that would mean some are impossible to construct for not all real numbers are constructible (for instance all real algebraic numbers of degree 3 over Q are not constructible). I am really not sure about this but the reasoning seems to work.

Originally Posted by JaneBennet

[FONT=Times New Roman]What you do is draw a circle of radius 12, mark 30 equally spaced points on the circumference, then draw a circle of radius 20 with any one of these points as centre. (Each knight can touch the lance of another knight up to 20 ft away if both knights use their lances.)

Here is a graph.

I don't see any Teaching going on here . . .

Originally Posted by Soroban

I don't see any Teaching going on here . . .

Why does that surprise me? Oh yes, I got it: none of us gave a complete, worked out, ready to hand in solution. Teaching is measured by learning. If the student has nothing to do, no learning can possibly take place. Learning is active not passive. Learning mathematics is not a specter activity; the learner must be evolved in the activity. Why don’t you review this entire thread? I think you would be surprised how much learning did go on. I, for one, learned a great deal about graphing the problem using a minimal edition of MathCad. Any student who uses such a CAS, would learn a great deal of mathematics in the simple act of modeling this problem as JaneBennet smartly suggested.