NUMB3RS Activity: Irregular Polygon Centroids. Episode: Burn Rate

Transcription

1 Techer Pge 1 NUMB3RS Activit: Irregulr Polgon Centroids Topic: Geoetr, Points of Concurrenc Grde Level: 9-10 Ojective: Students will e le to find the centroid of irregulr polgons. Tie: 0 inutes Mterils: TI-83 Plus/TI-84 Plus grphing clcultor Introduction In Burn Rte, when il os kill series of seeingl unrelted people, Don sks Chrlie to help uncover the source nd the link tht connects the people. Don s te hs deterined the loctions where the letter os were iled, where the envelopes were purchsed, hrdwre store where soe coponents were ought, etc. Using geo-profiling, Chrlie is le to find the likel strting point where the oer set out to u his o terils. Geo-profiling is n investigtive technique used lw enforceent tht uses the loctions of connected cries to deterine the ost prole re of offender residence. In this ctivit, students will do their own geo-profiling finding the centroid of polgon where the vertices of the polgon will represent the loctions tht Don s te hs discovered. An lgeric ethod will e deterined for finding the centroid of weighted nd non-weighted dt points. The etension of this ctivit detils ethod for finding the centroid of non-polgonl region using clculus. Discuss with Students Students should e le to construct edin using copss or pper folding. To id in the clcultions for finding weighted centroids, downlod the clcultor progr CENTROID going to the We site nd serching for This progr will e utilized in Question 6. Student Pge Answers: 1.. (3, 1.7) 007 Tes Instruents Incorported

2 Techer Pge 3. Centroid nswers vr. 4. (5.4, 6.1) 5. (4.7, 6.5) 6. (.9, 5.1) Etension Pge Answers: 1. = ( + + ) d = M = ( + + ) d =.5 1 ( ) 10.8 M = + d =. M.5 = = = M 0.8 = = = The Centroid is pproitel (0.5,.4) 007 Tes Instruents Incorported

3 Student Pge 1 Ne: Dte: NUMB3RS Activit: Irregulr Polgon Centroids In Burn Rte, when il os kill series of seeingl unrelted people, Don sks Chrlie to help uncover the source nd the link tht connects the people. Don s te hs deterined the loctions where the letter os were iled, where the envelopes were purchsed, hrdwre store where soe coponents were ought, etc. Using geo-profiling, Chrlie is le to find the likel strting point where the oer set out to u his o terils. This strting point will e the centroid of the polgon deterined the loctions found Don s te. Reeer tht centroid is the lncing point of the polgon. In other words, if the polgon were to e plced on the tip of the pin t the centroid, it would e perfectl lnced. 1. Suppose Don hs deterined tht the loctions re (0, 0), (6, 0) nd (3, 5.) when lid out on p. Plot the points nd find the loction of the centroid finding the intersection of the three edins. While this works ver well, it is dependnt upon geoetric constructions which cn ecoe proletic when deling with p coordintes given Don s te. An lgeric ethod for solving the previousl stted prole is to find the centroid using the forul , Find the centroid using this forul to confir its ccurc. This ethod cn not onl e used for tringles, ut for irregulr shped polgons, which is wht the dt points supplied Don s te will result in when grphed. 3. Suppose Don s te hs deterined the oer used the following loctions (1, 5), (3, 10), (, 7), (6, ), (7, 9), (10, 3) nd (9, 7). Plot these points on the grid elow nd estite the centroid. 007 Tes Instruents Incorported

4 Student Pge To deterine the ect loction of the centroid, the originl forul will e etended such tht: = verge of -coordintes = verge of -coordintes Let i e the -coordinte in the ith dt point, i e the -coordinte in the ith dt point, nd n e the totl nuer of dt points. Then the centroid of the dt cn e n n surized the epressions in the ordered pir i i. i= 1 i= 1, n n 4. Clculte the centroid using this new forul nd see how close our guess ws. In finding the loction of the oer, Chrlie decides to weight certin dt points ore thn others ecuse of the iportnce of the to the oer. Suppose the dt points ove were weighted s follows: (1, 5) weight (10, 3) weight 1 (3, 10) weight 1 (8, 9) weight (, 7) weight 3 (9, 7) weight 1 (6, ) weight 1 To copute the weighted centroid, slight odifiction to the forul will e needed. Ech dt point ust e ultiplied its weight efore suing, nd ech su ust e divided the totl of the weights ttched to ll the dt points. n n i i i i i 1 i 1 This cn e surized the epression = =,. n n i i i= 1 i= 1 5. Clculte the weighted centroid. 6. Using the clcultor progr supplied our techer, copute the weighted centroid for the following dt points: (1, 3) weight 3 (4, 8) weight 1 (8, 3) weight 1 (1, 1.3) weight (0, 8) weight (0, 6) weight (5, 1) weight 1 (9, 8) weight 1 (3.5, 6) weight 1 (9, 11) weight Tes Instruents Incorported

5 The gol of this ctivit is to give our students short nd siple snpshot into ver etensive theticl topic. TI nd NCTM encourge ou nd our students to lern ore out this topic using the etensions provided elow nd through our own independent reserch. Etension: Finding the Centroid of Sooth Curve The centroid (, ) for the region ounded f() nd g() is defined s M =, where M = [( f ) g( )] d, = [( f ) g( )] d. M = nd f( ) + g( ) M = [( f ) g ( )] d, nd To find the centroid of the region ounded the two equtions f() = nd g() = 4, we notice fro the grph tht the intersect t ( 1, 3) nd (, 0) Find, M nd M for f() nd g() given ove. = [ ( 4)] d = M = [ ( 4)] d = + 4 [ ( 4)] M = d =. Using the vlues ou found ove, wht re nd? M = = M = = 007 Tes Instruents Incorported