Cutting a Semicircle into 2 Parts

**yeah:)** · September 19th 2010, 01:43 PM

How do you solve this?

**Wilmer** · September 19th 2010, 06:18 PM

The horizontal line will create a segment; go here:
Circle Sector and Segment

**pickslides** · September 19th 2010, 08:54 PM

Not sure if this is the simplest way but you require half the area of a semi circle to be $\frac{1}{4}\times \pi\times r^2$ to equal the segment part of $\frac{1}{2}\times (\theta -\sin \theta) r^2$ giving

$\frac{1}{4}\times \pi\times r^2=\frac{1}{2}\times (\theta -\sin \theta) r^2$

$\frac{1}{4}\times \pi=\frac{1}{2}\times (\theta -\sin \theta)$

$\frac{\pi}{2}= \theta -\sin \theta$

**yeah:)** · September 20th 2010, 08:27 AM

Originally Posted by pickslides

Not sure if this is the simplest way but you require half the area of a semi circle to be $\frac{1}{4}\times \pi\times r^2$ to equal the segment part of $\frac{1}{2}\times (\theta -\sin \theta) r^2$ giving

$\frac{1}{4}\times \pi\times r^2=\frac{1}{2}\times (\theta -\sin \theta) r^2$

$\frac{1}{4}\times \pi=\frac{1}{2}\times (\theta -\sin \theta)$

$\frac{\pi}{2}= \theta -\sin \theta$

What is the most simple way of going from this equation (i.e. what is the easiest way of solving it, obtaining an accurate, numerical value)? How do you then find what the problem is looking for?

**emakarov** · September 20th 2010, 02:03 PM

What is the most simple way of going from this equation (i.e. what is the easiest way of solving it, obtaining an accurate, numerical value)?

Incidentally, there is a similar question in the Trigonometry forum, but it does not offer a precise solution (yet).

**yeah:)** · September 20th 2010, 02:20 PM

Originally Posted by emakarov

Incidentally, there is a similar question in the Trigonometry forum, but it does not offer a precise solution (yet).

So, would anyone be able to propose a clever method of solving this problem?

**pickslides** · September 20th 2010, 02:24 PM

Originally Posted by yeah:)

So, would anyone be able to propose a clever method of solving this problem?

We are solving for the angle $\theta$ which can be done numerically. Do this using a spreadsheet or graphical calculator.

**emakarov** · September 20th 2010, 02:36 PM

It is not clear to me what the phrase "find... the line" means in the original question. It is possible that the distance of the bisecting line from the diameter is a nice number expression built from natural numbers, $\pi$ , the four arithmetic operations, radicals, and trigonometric functions. However, there may not be such an expression since the number of reals is vastly greater than the number of expressions (even though both are infinite). So, sometimes the description "the number $x$ such that $x-\sin x=\pi/2$ " is as good as one can get. I would ask the instructor to clarify what kind of answer is expected. I believe that in this case it is not something like asking for a hint.

**pickslides** · September 20th 2010, 02:53 PM

Agreed, it seems more practical to ask what is the reduction in the radius such that the semicircle is half the area given the horizontal cut.

**yeah:)** · September 20th 2010, 03:26 PM

Originally Posted by pickslides

Agreed, it seems more practical to ask what is the reduction in the radius such that the semicircle is half the area given the horizontal cut.

The answer expected is the point down the distance of the radius of the circle, through which it would be possible to draw a horizontal line, in order to cut the semicircle into 2 parts exactly. What is the most simple way of obtaining an accurate numerical value for this point?

**pickslides** · September 20th 2010, 04:31 PM

This is answered in post #7.

You just need to then find the relationship between r and $\theta$

**Wilmer** · September 20th 2010, 05:38 PM

Originally Posted by yeah:)

What is the most simple way of obtaining an accurate numerical value for this point?

Define "simple". "Simple" to you may be "complicated" to me ... or the other way around!

**yeah:)** · September 20th 2010, 10:06 PM

Originally Posted by Wilmer

Define "simple". "Simple" to you may be "complicated" to me ... or the other way around!

The main thing is that you explain it well to me - please show me, with explanation, how you would solve this problem.

**mr fantastic** · September 20th 2010, 10:16 PM

Originally Posted by yeah:)

The main thing is that you explain it well to me - please show me, with explanation, how you would solve this problem. For example, how would you do it with the Newton method, or any other iteration?

You have been given the required equation and shown where it comes from. You have been told to take a numerical approach in solving the equation (because an exact solution is not possible). As far as I can see, the question you originally asked has been completely answered.

It is up to you now to decide what to do. What numerical approaches have you been taught? Do you know how to use technology (such as a graphics calculator) to solve an equation? These issues (that were not flagged in the original post) do not belong in the Geometry subforum.

**yeah:)** · September 21st 2010, 12:38 PM

OK - thank you very much!

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