11. PYTHAGORAS THEOREM


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1 11. PYTHAGORAS THEOREM 111 Along the Nile Proofs of Pythgors theorem Finding sides nd ngles Semiirles Surds Chlking hndll ourt Pythgors prolems Designing toolox Squring the irle , MMster & Mithelmore 1
2 Ativity 11 1 Along the Nile Rmeses ws not hppy euse he ws left with smller re of lnd thn he hd efore. The ngle etween the knotted rope nd eh line of stones should hve een 90º. A rightngled tringle on the grid tht hs se of 1m nd height of 1m hs hypotenuse of 1.4m. One rightngled tringle in your tle hs hypotenuse tht is lso whole numer. The 3 side lengths of this tringle re 3m, 4m nd 5m. Yes. A rightngled tringle twie s long nd twie s high s this tringle lso hs whole numer side lengths. (This is expeted euse they re similr tringles). From the tle elow, you should find tht = 2 ie. The squre of the hypotenuse is equl to the sum of the squres of the other two sides , MMster & Mithelmore 2
3 Proofs of Pythgors Theorem Ativity 11 2 If Pythgors theorem is true, the re of the lrgest squre (the squre on the hypotenuse) is equl to the sum of the res of the other two squres. Yes. The tringles whih mke up the two smller squres n ll e found in the squre on the hypotenuse. The re of the squre spe remining is 2. In the other squre (hving the sme size) the remining spe is The remining spe must e the sme in eh squre euse the sme 4 tringles were psted in eh one, so = , MMster & Mithelmore 3
4 Totl re of shpe drwn: The side of the squre spe elow is . Totl re of shpe drwn: 2 The side of the squre spe elow is . The totl re of the top shpe is the sme s the totl re of the ottom shpe euse they re oth overed y 4 ongruent tringles leving spe of ( ) 2. Therefore = , MMster & Mithelmore 4
5 Finding sides nd ngles Ativity 11 3 When side length = 6m, = 8m nd = 4m, no, the ngle etween sides nd looks like rightngle. No When side length = 6m, = 8m nd = 6m, no, the ngle etween sides nd does not look like rightngle. No Try your other strw lengths for side. Wht length must side e for the ngle etween sides nd to look like rightngle? = 10m Yes = 2 If < 2, the ngle etween the side with length nd the side with length is > 90º. If > 2, the ngle etween the side with length nd the side with length is < 90º. In the tringle drwn, = 5m nd = 12m. = 13m. Yes. It looks like rightngled tringle. You tell without mesuring the ngle euse = 2. If ws the length of the shortest side (ie. = 5m) nd the side perpendiulr to it ws (ie. = 12m), no, The tringle elow is rightngled tringle. = 10.9m Yes. It looks like rightngled tringle. Yes = 2 = 5m nd = 12m. 5m 12m 2004, MMster & Mithelmore 5
6 In rightngled tringle, Wht is the length of in terms of nd : 2 = = Wht is the length of in terms of nd : 2 = 22 = 22 Wht is the length of in terms of nd : 2 = 22 = 22 Before nswering eh question elow, sketh nd lel the tringle (it does not hve to e to sle) so you know whih side is the hypotenuse. 1) A tringle hs one side 6m long, nd side perpendiulr to it is 8m long. The third side is 10m long. The re of the tringle is 24m 2. 2) A rightngled tringle hs se 12mm long. The longest side is 15mm long. The third side is 9mm long The re of the tringle is 54mm , MMster & Mithelmore 6
7 Ativity 11 4 Semiirles The length hypotenuse should e 5m long. Are of the semiirle on the se of the tringle = ½π2 2 = 2π Are of the semiirle on the height of the tringle = ½π(1½) 2 =1π Are of the semiirle on the hypotenuse = ½π(2½) 2 = 3π Are = 2π + 1π = 3π The re of the semiirle on the hypotenuse is equl to the re of the semiirles on the other two sides. Yes. The sme would e true for semiirles drwn on the 3 sides of other rightngled tringles euse the re of semiirle is diretly relted to the squre of the length of its dimeter. If it is rightngled tringle, the squres of the 3 dimeters of the semiirles re relted to eh other in the sme wy s the lengths of the 3 sides of the tringle. 2004, MMster & Mithelmore 7
8 Ativity 11 5 Surds 3 x x = = 13 = No is not surd euse when you multiply it y itself, the nswer is 3.14 nd 3.14 is not whole numer. No. π is not surd euse π 2 is not whole numer. The vlue of ( 13) 2 is 13. The smll rightngled tringle t the ottom of this pge: Length of hypotenuse = m = 2 m Now use this hypotenuse s the se of nother rightngled tringle with height of 1m. The height hs een drwn. Drw the hypotenuse. Length of new tringle s hypotenuse = ( 2) = 3 m Use the hypotenuse you drew s the se of nother rightngled tringle with height of 1m. Length of new tringle s hypotenuse = ( 3) = 4 m No. This nswer is not irrtionl numer euse 4 = 2. Mesure the length of this hypotenuse. Length = 2 m 2004, MMster & Mithelmore 8
9 Chlking hndll ourt Ativity 11 6 Yes. It is good ide to mesure the digonl lengths euse if the two digonls of qudrilterl re the sme length, you know it is retngle (ie. it hs opposite sides equl nd ll ngles re 90.) Digonl length of the ourt = mm = 516 mm Lengths to mesure Corret length Our ourt Another ourt Length of ourt 420mm Width of ourt 300mm Length of one digonl 516mm Length of the other digonl 516mm Length of one smll retngle 210mm Width of one smll retngle 150mm 2004, MMster & Mithelmore 9
10 Ativity 11 7 Pythgors prolems 600mm Length of eh wire = mm = 1082 mm 900mm guy rope tent pole 130m 180mm The gretest distne tht tent peg n e from the foot of tent pole is: mm = 125 mm This ssumes tht the ground is flt (ie. horizontl). 2004, MMster & Mithelmore 10
11 6.9m X 1.5m Y 3.1m On the digrm, the horizontl distne of the kite from the mn s hnd is lelled Y. The remining side of the rightngles tringle is lelled X. X = 3.1m  1.5m = 1.6m Y = m = 6.7m 3.6m 4.2m Distne the ot hs moved down = m = 2.2m 2004, MMster & Mithelmore 11
12 Ativity 11 8 Designing toolox Length of the digonl on the ottom of the toolox = m = 75m Length of the digonl on the rdord is 75mm. Minimum height of ox = mm = 28mm So the toolox should hve minimum height of 28m. 80mm 75mm digonl ross the ottom of the ox height of the ox 2004, MMster & Mithelmore 12
13 Ativity 11 9 Squring the irle If the irle hs dimeter of 10, eh side of the lrger squre is 10. The sides of the smller squre hve length x. x 2 + x 2 = 10 2 euse the dimeter of the smller squre is the so 2 x 2 = 100 hypotenuse of rightngled tringle tht hs two x 2 = 50 sides with length of x. x = 7.07 (This is the side length of the smller squre.) Side length of the middle squre = side length of lrger squre + side length of smller squre 2 = 8.54 (to 2 deiml ples) perimeter of the middle squre: 8.54 x 4 = perimeter of the irle: 10π = No. The perimeters of the squre nd the irle re not the sme. 2004, MMster & Mithelmore 13
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