Challenge Geometry Prob: Suppose segment XY is divided into millions of congruent...

onesun0000

Junior Member
Joined
Dec 18, 2018
Messages
83
I tried my very best but I can't figure out how to solve this. Please help.

Suppose a segment XY would be divided into millions of congruent segments and semicircles would be drawn on top each segment. What would the sum of the lengths of the arcs be?

Thank you
 
I tried my very best but I can't figure out how to solve this. Please help.

Suppose a segment XY would be divided into millions of congruent segments and semicircles would be drawn on top each segment. What would the sum of the lengths of the arcs be?

Thank you
Start small - and do some thought experiment.

Assume you have a line 100 cm long.

You draw a semicircle over it - what is the length of the arc?

Now cut the line into two pieces - and draw semicircles on top of each segment.

What would the sum of the lengths of the arcs be?

Now cut a 100 cm line into four pieces - and draw semicircles on top of each segment.

What would the sum of the lengths of the arcs be?

See any pattern.....
 
Suppose a segment XY would be divided into millions of congruent segments and semicircles would be drawn on top each segment. What would the sum of the lengths of the arcs be?

Suppose the length of XY is L, and you divide it into N congruent segments.

How long is each segment? What is the arc length of each semicircle? What is the sum of all of them?

By working with a variable and writing expressions, you can, in effect, carry out many examples all at once, and see what is always true.

If you still have trouble, please show us your answers to these questions, or whatever other work you do, so we can see where things might be going wrong.
 
The question behind the question, AKA "The question that is really being asked," is:

"Is the length of the arc proportional to the length of the diameter it subtends, or does the proportion change as the diameter gets longer?"

That is to say, the arc on a 1 cm line is a certain number of times the size of that 1 cm. Is the arc on a 2 cm line the same number of times the size of those 2 cm? Is the arc on a 100 cm line segment the same number of times as large as 100 cm? Are all of these similar? Do they have the same proportions?
 
What I've got is, the sum of the arc lengths would always be (π/2)(XY) no matter how many segments and semicircles were drawn within it.
 
The question behind the question, AKA "The question that is really being asked," is:

"Is the length of the arc proportional to the length of the diameter it subtends, or does the proportion change as the diameter gets longer?"

That is to say, the arc on a 1 cm line is a certain number of times the size of that 1 cm. Is the arc on a 2 cm line the same number of times the size of those 2 cm? Is the arc on a 100 cm line segment the same number of times as large as 100 cm? Are all of these similar? Do they have the same proportions?

I tried to solve again and I got [(pi)/2]XY no matter how many segments and semicircles are drawn.
 
What I've got is, the sum of the arc lengths would always be (π/2)(XY) no matter how many segments and semicircles were drawn within it.

That is correct. Each semicircle is pi/2 times the segment on which it is drawn, so the sum of the semicircles is pi/2 times the total segment, by the distributive property.
 
I tried to solve again and I got [(pi)/2]XY no matter how many segments and semicircles are drawn.

That was the "Aha!" moment that the question was meant to elicit. When procedures are taught by instruction, you learn the procedure, but are deprived of the opportunity to discover HOW to discover procedures. So a lot of math problems are just a flimsy pretext to get you to discover something for yourself.

In this case, it is that the ratio of the circumference to the diameter is a fixed ratio. In fact, it is the definition of pi itself. So applying it to a one meter ruler, or to both halves of a one meter ruler, or to all four quarters of a one meter ruler... it doesn't matter. It was just meant to give a real life, affectual understanding of similar shapes, fixed proportions, and the relationship that defines pi.

Most people who are "bad test takers" know the material. They just either miss the flimsy pretext, or look to hard for one that isn't there. It's more about understanding human nature than understanding math.
 
Top