Discussion about math, puzzles, games and fun. Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ π -¹ ² ³ °

You are not logged in.

- Topics: Active | Unanswered

Pages: **1**

**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 105,673

Hi;

This appeared in another thread.

Sector OAB is a quarter of a circle of radius 3. A circle is drawn inside this sector, tangent at three points. What is the radius of the inscribed circle? Express your answer in simplest radical form.

Let's see what geogebra can do.

1) Create point (0,3) and call it A. Create point (3,0) and call it B. Create point (0,0) and call it O.

2) Use the "Circular arc with center between two points," tool to create sector OAB.

3) Enter in the input bar f(x)=x.

4) Remove the x and y axes.

5) Draw line segments OA and OB.

6) Create a slider called c and min = 0 and max = 3. Create a point (f(c),f(c)) and call it C. This point will ride along f(x) when we adjust the slider.

7) Draw points (0,y(C)) and (x(C),0) they will be called D and E.

8) Draw line segments DC and CE. Check the algebra pane and the distances will be displayed.

9) Use the intersection tool to find the point on arc AB that intersects f(x). It will be called I.

10) Hide f(x) and draw line segment CI, your drawing should look like fig 1.

11) Enter in the input bar e + g - 2h. You should get in the number pane i = -1.171572875253812.

Remember to have set your options to 15 digits.

12) Select the slider and use the right arrow to click carefully until i becomes positive and then one click back with the left. The first stop will show the slider at 1.2

13) Adjust the slider to have an increment of .001 and then press the right arrow until i again just turns positive and the back it off by one. You should have i = -0.003093511065233.

14) Repeat this process until you get

i = -0.000000034374931

and

c = 1.242640680000005.

What we have here now is C being the center of the circle we want. c is the radius of that circle.

Your drawing should look like fig 2.

How can we get more from the number c = 1.242640680000005? That is easy, just take it over to Wolfram Alpha and enter this.

RootApproximant[1.242640680000005]

It returns

so that is the radius and we are done with geogebra.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.****No great discovery was ever made without a bold guess. **

Offline

**ElainaVW****Member**- Registered: 2013-04-29
- Posts: 553

I tried it and it worked. I am just learning the program.

That is to 8 digits of precision. The amount the program allows in the increment box. Is there a way to get an answer that is even better?

Offline

**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 105,673

Hi ElainaVW;

So nice to see you in here. In answer to your question there are two ways that I know of to increase precision.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.****No great discovery was ever made without a bold guess. **

Offline

**ElainaVW****Member**- Registered: 2013-04-29
- Posts: 553

Bobbym:

Can you show how?

I am a little confused on the new methods of verifying the correctness of answers obtained through empirical means such as these. JB and PB were over here giving a lecture that I attended and I am afraid most of it was over my head. EA also explained it but I did not get it. Can you give your explanation. See you later

Offline

**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 105,673

I have never met JB or PB, what are they like? I understand they are at NASA...

I am not sure that I can explain at the same level any of them could. The way I understand it when you do a PSLQ a high probability of correctness is implied. Exactly how, I do not know.

To get more precision you now take the analytical form and plug it into c. Immediately you will see agreement between all three line segments to 15 digits. With one stroke, we have doubled the precision and the certainty of the answer is also increased.

Thanks for looking at the geogebra drawing, this is one of my favorites.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.****No great discovery was ever made without a bold guess. **

Offline

Pages: **1**