Grade 7/8 Math Circles Geometric Arithmetic October 31, 2012

Transcription

1 Fculty of Mthemtics Wterloo, Ontrio N2L 3G1 Grde 7/8 Mth Circles Geometric Arithmetic Octoer 31, 2012 Centre for Eduction in Mthemtics nd Computing Ancient Greece hs given irth to some of the most importnt fetures of Western civiliztion, including democrcy nd the modern deductive method. But for ll of their mzing chievements, they lcked one relly importnt feture - n effective numer system. In fct, it is even more mzing tht they did ll tht they did WITHOUT numers. The Greeks understood how numers intercted... ut they didn t hve wy to write them using symols. Insted, they hd to write them out in long hnd. For exmple, the Greeks would write the lgeric prolem Solve 2x + 5 = 13 s Let five dded to twice n unknown quntity e equl to thirteen. Wht is the unknown quntity? Not very plesnt to red, right? Imgine hving to do this for ll of your mth prolems! Why ws this the cse? There re mny resons... one fctor is tht the Greeks often viewed numers not s representtions of n mount, or how mny/much, ut rther, s mesuring length. This ws evident in their pproch to geometry - geometry ws one of their wys of representing numers! The Greeks did rithmetic with geometry - which is wht we will e lerning tody!

2 The Pythgoren Theorem One proof of the Pythgoren Theorem involves the following configurtion: To prove the theorem, we need one importnt fct. For ny two tringles, if two corresponding sides re the sme length, nd the ngle etween the two sides re the sme in either tringle, then Both tringles hve the sme corresponding ANGLES. 2. Both tringles hve the sme corresponding SIDE LENGTHS 3. In other words, they re the sme tringle. Exmple 2

3 In order to prove the Pythgoren Theorem, we first need to show tht the four sided shpe in the center is ctully squre. This requires two steps: 1. We need to show the side lengths re ll equl. 2. We need to show tht the ngles in the qudrilterl re ll equl. The Proof 1. In the left digrm, wht is the re of the grey squre in terms of c? c 2 2. Imgine moving the tringles round until you get the digrm on the right: c c c c c c 3. Wht is the totl re of the drk region? c 2 4. Wht is the re of the smller drk squre? 2 5. Wht is the re of the lrger drk squre? 2 6. Wht cn you conclude? Write it out lgericlly = c 2 3

4 Something even more interesting... Notice tht = 1 = = 4 = = 9 = = 16 = = 25 = = 36 = = 49 = 7 2. n (2n 1) = n 2 Pttern: The SUM of consecutive ODD numers is perfect SQUARE. Algeric Proof Beyond your cpilities. Geometric Proof Proof y picture. Let 1 represent squre of re 1, 3 represent 3 squres of re 1, n represent n squres of re 1... etc

5 Some Necessry Skills Constructing Perpendiculr Bisectors nd Midpoints A B 1. Widen your compss so tht one end is t A nd the pencil end is t B. Drw circle. 2. Repet step 1, except put the pencil t A, the other end t B. 3. These two circles should intersect in two points on the pge. Mrk these points, then use the strightedge to connect them. Erse the circles. 4. The line you just mde should hve crossed your originl line. This is the perpendiculr isector. 5. Mrk the point tht your perpendiculr isector cuts your originl line s the point M. 6. M is midwy etween A nd B - it is the midpoint! 5

6 Constructing Perpendiculrs to n Outside Point P A B 1. Set your compss to hve center t A, pencil t P. Drw the circle. 2. Set your compss to hve center t B, pencil t P. Drw the circle. 3. There should e two points of intersection of the two circles you drew. Mrk them s S nd T, then drw line etween them. Erse the circles. 4. This is the perpendiculr line tht psses through P! 6

7 Constructing Perpendiculr Through n Endpoint A B 1. Extend the line AB with your strightedge, t lest doule the length. 2. Use your compss to drw circle with center t B nd rdius AB (so put your pencil t A). 3. Mrk the point of intersection of tht circle with the line. Cll tht point C. Erse the circles. This forms line segment, AC. 4. Plce the center of your compss t C, the pencil t A. Drw the circle. Repet, with the center t A nd pencil t C. Connect the two points like you did in the midpoint construction. Erse ll the circles ut KEEP the perpendiculr line. 5. The line tht you mde will pss through B, nd it is the perpendiculr through B. 7

8 Constructing Prllel Lines Through Point P A B 1. Using the necessry skill Constructing Perpendiculrs to n Outside Point, construct perpendiculr line to AB tht psses through P. Erse everything ut the perpendiculr line. 2. Mrk the point of intersection of your perpendiculr line nd AB s point Q. 3. Rotte your pge, if you wish. Using the line segment P Q, construct perpendiculr line to P Q tht psses through P (the endpoint) like you lerned efore. Erse ll unnecessry mrks when done. 4. Tht line is prllel to AB! 8

9 Mth without Numers! Multipliction O x y 9

10 1. Using wht you lerned to construct perpendiculrs through endpoints, crete perpendiculr through the point mrked O - mke it tll! Erse everything, EXCEPT this perpendiculr line. 2. Drw circle of rdius 1 unit (use the dshes) with center t O. Mrk the TOPMOST point of intersection of this line with the circle you just drew s point P. Erse the circle. 3. Drw line using strightedge through P nd x. 4. Drw circle of rdius y with center t O. Mrk the TOPMOST point of intersection of this circle nd the perpendiculr through O s point Q. Erse the circle. 5. Refer ck to your line P x. Extend this line upwrds nd to the left; mrk point out 4 cm to the left of P, cll it R. 6. Using the line segment RP nd the point Q, construct perpendiculr line to RP tht psses through Q (you ve lerned how to do this!). Mrk the point of intersection of this line with RP s the point S. (This is getting messy now - you should proly erse ll your circles!). 7. Using SQ s your line segment, drw perpendiculr line tht psses through Q (refer ck to the necessry skill - drwing perpendiculr through n endpoint). 8. Extend this line until it hits the line with the rrow. Mrk the point of intersection of the line you just drew with the horizontl line s point xy. 9. Mesure the lengths of x, y, nd xy using the dshes. How long is xy? Wht is the reltionship? 10

11 Division A B 11

12 Division At every step, erse ll unnecessry mrkings to keep it net! 1. Using wht you lerned in your necessry skills (perpendiculr through n endpoint), drw perpendiculr through A nd nother one through B. 2. On the top hlf of the perpendiculr through A, pick point roughly 2 cm (I m serious, do NOT pick more thn 5 cm) ove A nd mrk it s P Put the center of your compss t P 1, pencil t A, nd drw circle. Mrk the other point of intersection with the perpendiculr s P Put the center of your compss t P 2, pencil t P 1, nd drw circle. Mrk the other point of intersection with the perpendiculr s P Repet steps 3 nd 4 until you hve mde point P 4. These points should e eqully spced. 6. Use the necessry skills you ve lerned to drw line prllel to AB tht psses through P 1 (if you know how to chet, go hed here, it s not tht importnt for this step). 7. Mrk the point of intersection of this line with the perpendiculr through B s point Q. 8. Drw the circle with rdius BQ, with center t B. Mrk the other point of intersection of this circle with the perpendiculr through B s Q Repet steps 2-5, except using Q 1, Q 2, etc. until you ve mde Q Join P 1 to Q 4, P 2 to Q 3, P 3 to Q 2, nd P 4 to Q 1, with strightedge. Mrk the points where these lines intersect the line AB. 11. Those points you mrked ove divide AB into segments. Mesure the length of the entire segment, nd then mesure the lengths of ech smll segment - wht do you notice? Wht rithmetic opertion hve you performed? 12

13 Squre Root (x x) O 1 cm A x B 1. Construct the midpoint of the line OB. Mrk this point s D. Erse everything else. 2. Construct semi-circle of rdius OD with center t D. 3. Drw perpendiculr through A. Mrk the point of intersection of the perpendiculr with the semi-circle s the point P. 4. Mesure the length of AB. 5. Mesure the length of AP. 6. The length of AP is the squre root of x! 13

14 Squring (x x 2 ) O 1 cm A x 1. Use the segment OA s your line segment. Drw perpendiculr to OA pssing through its endpoint, A. 2. Plce the center of your compss t A, pencil t x, nd drw circle. Mrk the point of intersection of this circle with the perpendiculr you just constructed, s the point P. Erse the circle. 3. Drw line from O to P. 4. Find the midpoint of the line segment OP ; mrk this point s D. Do not erse the perpendiculr isector. 5. Using the perpendiculr isector your just mde in the previous step, extend it until it crosses the rrowed line. Lel the point of intersection s C. 6. Construct circle of rdius CO with center t C. Lel the point of intersection of this circle with the rrowed line, nd mrk it s S. 7. Mesure the length of Ax using the dshes (distnce etween two dshes is 1 unit). Mesure the length of AS. 8. Wht is the reltionship etween the lengths you just mesured? You hve just squred the length of Ax! 14

15 Some Cooler Constructions... Pppus Theorem P A B Instructions 1. Construct the prllel line to AB tht psses through P (refer ck to necessry skills) 2. Mrk three rndom points on the line AB (etween A nd B!). Cll these X, Y, nd Z, from left to right. 3. Mrk three rndom points on the line through P. Cll them U, V, W, from left to right. 4. Construct the following lines: UY, UZ; V X, V Z; W X, W Y. () Lel the intersection of UY nd V X s point R. () Lel the intersection of UZ nd XW s point S. (c) Lel the intersection of W Y nd V Z s point T. 5. Connect points R, S, nd T. (Isn t it mzing tht they lie on stright line?) Try with different points! You will ALWAYS get the sme result. 15

16 Tngents to Circle From Point O P 1. Connect O to P with strightedge, forming segment OP. 2. Find the midpoint of the segment OP, cll it Q. 3. Use your compss to construct the circle of rdius QP with center t Q. 4. This circle will intersect your originl circle in two points. Mrk these points s S nd T. 5. Drw line from P to S, nd line from P to T. These re the tngent lines to the circle tht pss through P. 16

17 Md Skills - Three Tngent Circles O 1. Pick ny point P on the circle s circumference. 2. Drw the line OP, mke it firly long. 3. Drw circle with rdius OP, with center t P. Mrk the point of intersection of the circle with the line you mde in Step 2 s Q. 4. Drw the circle of rdius QP with center t Q. This circle is tngent to the first! 5. Drw line perpendiculr to OQ tht psses through P. Cll this line L. Extend this line in oth directions. 6. Construct the two tngents from Q to the first circle (you lerned this in the previous step). Pick one of the points of tngency nd cll it S. 7. Drw the line OS until it psses through L. Mrk this point s R. 8. Drw the circle with rdius RS with center t R. This circle will e tngent to the other two circles! 17

18 Circle within Tringle... within Circle! A B 1. Drw circle with rdius AB with center t A. 2. Drw circle with rdius BA with center t B. 3. Mrk the topmost point of intersection s C. Drw the tringle formed y these three points. This is n equilterl tringle! 4. Erse everything in your drwing except for the equilterl tringle. 5. Find the midpoints of ech of the sides of the tringle (3 totl. (Refer ck to necessry skills) 6. From ech point, drw line to the vertex of the tringle tht is directly opposite of it. You should hve three lines tht meet t one point. Cll this point O. 7. Pick one of the midpoints. Put the center of your compss t O, the pencil t the midpoint. Rotte your compss - wht hppens? 8. You hve just mde circle inside tringle which is inside two circles! 18

19 Chllenge Investigtion: ( + ) 2 =? An importnt lger fct tht you will lern in high school is tht if you hve two whole numers nd, then ( + ) 2 = (Algeric) Proof: ( + ) 2 = ( + )( + ) Let x = ( + ). Then ( + ) 2 = ( + )( + ) = x( + ) = x + x (Distriutive Property) = ( + )() + ( + )() (Sustitute x = ( + )) = [ + ] + [ + ] (Distriutive Property gin) = = ( = ) (Fill in the remining steps of the lgeric proof.) Cn you prove the sme result y using geometricl rguments nd the following digrm? 19