RECTIFYING THE CIRCUMFERENCE WITH GEOGEBRA

Transcription

1 ECTIFYING THE CICUMFEENCE WITH GEOGEBA A. Matín Dinnbie, G. Matín González and Anthony C.M. O 1 Intoducction The elation between the cicumfeence and the adius of a cicle is one of the most impotant concepts explained duing the fist yeas in the academic path of a math student. This elation is not a tivial concept and duing centuies has fascinated and get the attention of mathematicians and scientists fom the ancient Egypt to the moden supecomputes whose powe has been tested by calculating digits of. Achimedes ceated an algoithm fo calculating it in his book Measuement of the cicle but it was Leonad Eule who in 1737 intoduced the use of the Geek lette to efe to the numbe we know as pi. 1. Intoducing the elation to Elementay and Seconday students fo the fist time can be a complex task. As a matte of fact the compehension of pi took centuies to mankind but nowadays the pogams of study spend shot time explaining it. Nowadays new technologies can be extemely useful in explaining this elation and Geogeba is one of the mathematic digital tools that stands out among the othes fo this pupose. The constuction of the animation explained in this aticle povides a valuable didactic esouce to facilitate the students compehension of the elation between the cicumfeence and the diamete of a cicle. In 1882, the Geman mathematician Fedinand Lindemann poved that is a tanscendental numbe, meaning it is not a oot of any polynomial with ational coe cients. The most impotant consequence of this is the impossibility of solving one of the thee unsolved poblems of ancient Geece: squaing the cicle, also known as the quadatue of a cicle. The tanscendence of pi implies the impossibility of cicling the squae, as well as squaing the cicle by compass and staightedge. A cicle can be squaed but not exclusively by using compass and staightedge. 2 The impossibility of squaing the cicle implies the impossibility of ectifying the cicumfeence, as we ae going to explain a segment of length equivalent to the length of a cicumfeence can t be constucted by compass and staightedge. The impossibility of solving the othe two poblems of ancient Geece, tisecting an angle and doubling a cube, is deduced by othe methods. 1.1 The squae of the cicle and the ectifying of the cicumfeence The impossibility of squaing the cicle implies the impossibility of ectifying the cicumfeence Conside the following geometic pocess:

2 1. Constuct a line segment of length +, and a semicicle with cente in its middle point (It is assumed that > 0) 2. Constuct a line segment pependicula to the pevious segment with oigin in and end in the intesection with the semicicumfeence. 3. The dawing shows a ight tiangle. The angle coesponding to the A vetex is a ight angle. A c h b B C As the line segment h is the height of the tiangle using the geometic mean theoem we have that: h 2 = = 2 =) h = p 4. Constucting a squae of side l = p l = p ectifying the cicumfeence with Geogeba 2

3 5. The suface of the squae would be S = ( p ) 2 = 2. Theefoe we have constucted a squae whose suface is the same than a cicle of adius. It seems that we achieved ou goal: the squae of the cicle. Whee is the poblem? Have we used stictly compass and staightedge to daw the pevious constuction? Lindemann did not poved that the squae of the cicle was impossible, he poved that it is not possible by using only compass and staightedge. If in the pevious constuction the dawing of a line segment of length using staightedge and compass was possible, the squaing of the cicle would be possible. Knowing that the squaing of the cicle is not possible we can conclude that the ectifying of the cicumfeence is not possible. 1.2 ectifying the cicumfeence with Geogeba Geogeba is based on the staightedge and the compass. Theefoe it is not possible with this softwae to solve the poblem of ectifying the cicumfeence. Howeve, as we ae going to show it is possible to obtain an acceptable appoximation to the ectify of the cicumfeence. 2 Explanation of the poblem and its elements 2.1 The final constuction The pupose of this document is the explanation of the pocess of unfolding a cicumfeence to become a line segment on the OX axis using Geogeba. The final constuction can be found at and it is inspied by a constuction of the Geogeba institute of HongKong by Anthony C.M. O that can be found hee: https: // Figue 1: Animation In ode to undestand the animation a sequence is showed in Figue Key element definition Let s define some key elements of the constuction on figue 2 1. es the adius of the cicumfeence whose lenght is going to be unfold becoming a line segment. The length of the line segment is the cicumfeence which is going to be ectify. ectifying the cicumfeence with Geogeba 3

4 A B Figue 2: Key Elements 2. is the adius of the ac of cicle that ceates the animation. It is an ac of a fixed lenght of a cicumfeence with a vaiable adius. 3. The point A is the cente of the cicumfeence of adius, which moves along the positive X-axis. 4. The point B is the endpoint of the ac of cicle that is going to be unfolded 3 Analyisis of the animation elements 3.1 The point A A slide in Geogeba is a contol to modify a specific value by the use. In this paticula case we will build a slide fo the m vaiable that would change fom 0 to 1 allowing to daw the point A giving the coodinates (0, ) to it. As we said befoe, the point A moves away fom the oigin of the coodinate system on the positive Y-axis as the cicumfeence is being unfold. The point moves fom the cente of the cicle that is going to be ectify ((0,) coodinates) moving away fom the oigin of the coodinate system as the point moves along the Y-axis, theefoe its fist coodinate would be always ceo, moevove > 0 and appoaches +1. It can be seen that when the ac of cicle had been completely unfolded, the value of m will be 1 and theefoe the value of the second coodinate of point A will be undefined as we defined it as ). At any point befoe eaching the hoizontal position, the point A would be on the positive Y-axis As we said befoe a way to epesent this situation is to ceate a slide in Geogeba called m. A(0, ) Point A coodinates ectifying the cicumfeence with Geogeba 4

5 Notice that lim =+1. Funtion f (x) = gows appoaching +1 as m appoaches 1 fom the left. The slide will do the point A to move along the Y-axis fom coodinate (0,) to (0,+1) The behaviou of the point A can be undestood bette looking figue 3 Figue 3: The point A moves fom (0,) along the positive Y-axis 3.2 Point B The point B is the endpoint of the ac of cicle that is going to be ectified. In othe wods, the cicumfeence becomes an ac whose endpoints ae the oigin of the coodinate system (0,0) and the point B(x 1, x 2 ) The coodinates of point B ae defined in figue 4. In this figue can be seen the ight tiangle ABC and the angle whose sin is sin( ) = x 1. Knowing that x 2 is the value of the adjacent side to in that tiangle, we have that cos( ) = x The angle In the figue 5 it can bee seen the cicumfeence to be ectified with adius, and the ac in which is tansfomed. This ac has a adius whee one of the endpoints is the point A and the othe endpoint is (0,0). Being the cental angles popotional to the length of the coespondent ac, we will have (notice that the angles ae measued in adians): ectifying the cicumfeence with Geogeba 5

6 = 10.8 A C x 1 B x 2 = Figue 4: Point B coodinates A B Figue 5: atio of adiuses 2 = 2 2 Solving fo we obtain that = 2 measued in adians. emembe that when we talked about the point A we defined it with the coodinates (0, ). It means that the adius of the big cicumfeence with cente in A is. By substituting in the pevious expession: = 2 = 2 (). = 2 = 2 = 2 = 2 (). Volviendo a las coodenadas del punto B: The coodinate x 1 can be calculated using the expession sin( ) = x 1 sin( ) = x 1 =) x 1 = sin( ) = sin(2 ()) = sin(2 ()) x 1 = sin(2 ()) The coodinate x 2 can be calculated solving the expession cos( ) = x 2 ectifying the cicumfeence with Geogeba 6

7 cos( ) = x 2 =) x 2 = cos( ) = (1 cos( )) = (1 cos(2 ())) x 2 = (1 cos(2 ())) 3.4 Analytic poof of the elation between the coodinate x 1 and the length of the cicumfeence when m! 1 It can be poved that when m! 1 the coodinate x 1 becomes the length of the cicumfeence. sin(x) sin(x) emembeing that lim = lim = 1, and knowing that x is measued in adians. x!0 x x!0 + x We have that: (Be awae that if m! 1 then ()! 0) lim sin(2 (1 m)) apple 0 = 0 = 2 lim sin(2 ()) 2 () = 2 In the same way it is tue fo m! 1 +, theefoe: lim sin(2 (1 m)) = 2 Fo the calculation of this limit L Hôpital s ule can be used, but that would assume the use of deivatives is known. 3.5 In ode to calculate the coodinate x 2 it can be poved that it appoaches 0 when m! 1 eithe fom ight o left. That implies that the ectified cicumfeence will be on the positive X-axis lim apple (1 cos(2 ())) 0 = 0 = (1 ) lim sin( ()) 2 lim (2sin h i 2 2 ())) 2 sin( ()) 1 = = = 2 lim sin( ()) () sin( ()) 1 = = 0 The identity 1 cos(x) = 2sin 2 ( x 2 ) has been used in (1 ) The same esult will be obtained fo m! 1, so (1 cos(2 ())) lim = 0 ectifying the cicumfeence with Geogeba 7

8 4 Concluding emaks The point A can not exist when m = 1. Theefoe we can wonde why when m = 1 the line segment is being dawn. The answe is that when m = 1 the coodinates of the point A ae not defined and we foced the softwae to daw a line segment of an appoximate length 2. This can be done by using the visibility condition of a Geogeba object, that can be found in the advanced section of the popety dialog. In that section we can include conditions to show an object and in this paticula case the condition will be m = 1. In the animation we will add also thee whole cicles and the popotional pat of a fouth one with the same condition to show the elation in a gaphic way. 5 elated constuctions The pevious wok explained hee can be used to build othe constuctions with some didactic value. The next constuction can help in the explanation and compehension when calculating the aea of a cicumfeence. The Geogeba file can found in the link geogeba.og/m/bmdpmwez and it shows a cicumfeence of adius and the constuction of a tiangle with the same suface, whose base is and whose height is. The same explanation we did in the fist animation should be done hee because as we explained the cicumfeence can t be ectified by compass and staightedge. The animation build in this section uses the pevious analysis of this document but daws multiple concentic cicumfeences that ae unfolded until being completely hoizontal. Each cicumfeence has been divided into two acs of the same length as can be seen in the figue 6. The Geogeba file has been designed using a n slide that contols the numbe of cicumfeences allowing to incease the numbe of them as it is shown in the figue 7 In the figue 8 can be seen the final tiangle with an height equal to the adius of the cicumfeence and whose base is 2 3. Theefoe its aea can be calculated as: A = base height 2 = 2 2 = 2 3 As it was said befoe the base is not eally 2 long as the cicumfeence can t be ectified by compass and staightedge. ectifying the cicumfeence with Geogeba 8

9 Figue 6: elated constuctions ectifying the cicumfeence with Geogeba 9

10 Figue 7: Constuction inceasing the numbe of cicumfeences 2 Figue 8: Aea of the tiangle This document was ceated with LATEX by Adia n Matı n Dinnbie y Gema n Matı n Gonza lez is licensed unde a license Ceative Commons econocimiento-nocomecial-compatiigual 4.0 Intenacional License ectifying the cicumfeence with Geogeba 10