Pythagoras Theorem PYTHAGORAS THEOREM.

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1 Pthgors Theorem PYTHAGORAS THEOREM

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3 How oes it work? Solutions Pthgors Theorem Pge 3 questions Right-ngle tringles D E x z Hotenuse is sie: F Hotenuse is sie: DF Q k j l Hotenuse is sie: k P R Hotenuse is sie: PQ Nme the hotenuse for eh of these l rwn tringles. M L N Hotenuse is sie: Hotenuse is sie: MN Mthletis Pssort 3P Lerning I SERIES TOPIC

4 How oes it work? Solutions Pthgors Theorem Pge 5 questions Squres n right-ngle tringles 3 units Are 5 units # 5 units 5 units 3 Are units # units 44 units 5 units Are 3 3 units # 3 units 69 units Are + Are 5 units + 44 units 69 units units Are 3 Are 6 units # 6 units 36 units 3 Are 8 units # 8 units 64 units 0 units 6 units Are 3 0 units # 0 units 00 units 8 units Are + Are 36 units + 64 units 00 units Are 3 I SERIES TOPIC Mthletis Pssort 3P Lerning

5 How oes it work? Solutions Pthgors Theorem Pge questions Pthgors Theorem for right-ngle tringles Right-ngle Not right-ngle Right-ngle Not right-ngle Right-ngle Not right-ngle Right-ngle Not right-ngle e f Right-ngle Not right-ngle Right-ngle Not right-ngle Mthletis Pssort 3P Lerning I SERIES TOPIC 3

6 How oes it work? Solutions Pthgors Theorem Pge 8 questions Pthgors Theorem for right-ngle tringles A J K I 4.5 H J B M K 9 A 5 0 C ! 56 L 48 N ! H 5 G The right-ngle tringles re: ΔAJK, ΔHIJ, ΔGHK 4 I SERIES TOPIC Mthletis Pssort 3P Lerning

7 How oes it work? Solutions Pthgors Theorem Pge 8 questions Pthgors Theorem for right-ngle tringles 3 Ern n wesome ssort with this one! Nme ll the right-ngle tringles in this imge n mrk where the right-ngles re with the orret smol. R 65 S The right-ngle tringles re: 5 T 380 ΔPUV ΔQRU P 5 Q 5 6 U ΔRSU ΔSTU V Pge 9 questions Pthgors Theorem for right-ngle tringles Assuming the sle of the ge is the sme s the originl rint, the mesurements shoul e s follows: NOTE: if not the sme sle, the sme reltionshi etween our mesurements shoul work mm mm mm mm mm 56 4 mm mm 5 36 mm mm mm 96 mm mm mm 8 40 mm mm mm mm mm Mthletis Pssort 3P Lerning I SERIES TOPIC 5

8 Where oes it work? Solutions Pthgors Theorem Pge questions Clulting the length of the hotenuse g g g g 89 0 g Pge questions Clulting the length of the hotenuse 3 h h h h 3. in ext squre root form n + 35 n n n 389 in ext squre root form 4 ( 0 units) + ( 9 units) ( 59. units) + ( 34. units) 00units + 8units units units 8units units 8 units units units units units to eiml les. 68. units to eiml les 6 I SERIES TOPIC Mthletis Pssort 3P Lerning

9 Where oes it work? Solutions Pthgors Theorem Pge 3 questions Clulting the length of the hotenuse 5 Stge Stge Stge m 55m m m to eiml les The totl length of the 3 stge flight th m m Pge 5 questions Clulting the length of short sie j 0-56 (8. units) -(8 units) j units -34 units j units j units j 4 9. units Mthletis Pssort 3P Lerning I SERIES TOPIC

10 Where oes it work? Solutions Pthgors Theorem Pge 6 questions Clulting the length of short sie 3 - w w w in ext squre root form w 65 in ext squre root form x x x x x to eiml oint x to eiml oint 8 I SERIES TOPIC Mthletis Pssort 3P Lerning

11 Where oes it work? Solutions Pthgors Theorem Pge questions Comintion of hotenuse n short sie lultions The seil nme given right-ngle tringle whih is extl one hlf of n equilterl tringle: H E M I E Q tringle I M 4. E Q e E 6 6. h H 30 g 60 6 Mthletis Pssort 3P Lerning I SERIES TOPIC 9

12 Where oes it work? Solutions Pthgors Theorem Pge 9 questions Alitions of Pthgors Theorem x (3 m) -( m) x 69m -44 m 3 m x m x 5m x 5 m x 5m 4 m ut (4 m) + (34 m) 34 m ut ut ut 64 m + 56 m 90 m ut 90 m ut m ut. 54 to nerest whole m 3. km Strt (i) (. km) + (3.9 km).89km + 5.km 3.9 km 8.km 8. km km Finish. 4.5km to eiml oints (ii) To voi the swm, Mil wlke 3.9 km +. km 5.6 km Mil wlke further 5.6 km km..35 km 0 I SERIES TOPIC Mthletis Pssort 3P Lerning

13 Where oes it work? Solutions Pthgors Theorem Pge 0 questions Alitions of Pthgors Theorem 4.6 m m (i) se se ( m) -(.6 m) 89m -6.6 m se 8.4 m se se se 84. m 6.8 m (ii) Are (se # height) ' Are Are Are ( 6. 8m 6. m) 43.68m '. 84 m # ' 5 C AB (8 m) + (3.3 m) AB 34m m B 3.3 m 54.4 m AB AB AB m m 8.3m A 8 m Digrm not rwn to sle. BC (54.4 m) + (3.3 m) BC m m BC 90.5m BC m BC 54.5m Distne roun wll 9m Distne AC AB + BC 8.3m+ 54.5m.8 m Shortest th Mthletis Pssort 3P Lerning I SERIES TOPIC

14 Where oes it work? Solutions Pthgors Theorem Pge 0 questions Alitions of Pthgors Theorem 6 m (i) ( m- 3 m) + (0 m) (35 m) + (0 m) 0 m 5 m m 565 m 3 m 565 m 5 m (ii) Perimeter of the trezium m+ 0 m+ 3m+ 5 m 554 m Pge questions Alitions of Pthgors Theorem Y WY YZ -WZ WX XZ -WZ XY WY -WX WY 3969 WX 900 X 65 WY 63 WX W 6 I SERIES TOPIC Mthletis Pssort 3P Lerning

15 Where oes it work? Solutions Pthgors Theorem Pge questions Alitions of Pthgors Theorem 8 Clulte the length of the le suort BD on the rne iture elow if CD 9.5 m, AB 6 m n BC 8.5 m 8.5 m C B 9.5 m 6 m D A AC BC -AB AD AC -DC BD AD + AB m AC BD 00 AC. 5 m BD 0 m Mthletis Pssort 3P Lerning I SERIES TOPIC 3

16 Wht else n ou o? Solutions Pthgors Theorem Pge 3 questions Pthgoren tris ", 6, 0, 0, 4, 6 sst! Note tht the re written in orer of size. ", ", 35, 3, " 9, 40, 4, 40 Show whether these sets of ositive integers form Pthgoren tri or not. ", 4, 5, " 4, 48, 50, ", 34, 36, + 4 5? ? ? ? ? ? 300! 96 Yes No Yes No Yes No " 5, 36, 39, e " 6, 60, 63, f ", 30, 3, ? ? ? 3856! 3969? ? ? 044! 96 Yes No Yes No Yes No 4 I SERIES TOPIC Mthletis Pssort 3P Lerning

17 Wht else n ou o? Solutions Pthgors Theorem Pge 5 questions Euli s formul for Pthgoren tris q - q q + q Tri # # { 3, 4, 5 } 3-8 # 3 # { 6, 8, 0 } 5 - # 5 # { 0,, 9 } # # { 3, 84, 85 } - 3 # # { 66,, 30 } - 8 # # {, 56, 65 } (i) Fin Pthgoren tri in whih n - q is equl to 33 - q 33 - q q q 6 q q 4 q # # q Pthgoren tri is { 33, 56, 65 } (ii) Fin Pthgoren tri in whih q 5 n + q is equl to 6 - q q # 6 # q 6-5 Pthgoren tri is {, 60, 6 } Mthletis Pssort 3P Lerning I SERIES TOPIC 5

18 Wht else n ou o? Solutions Pthgors Theorem Pge 6 questions Pthgoren tris 3 Fin grou of three integers tht inlues the numer 4 n forms Pthgoren tri. { - q, q, + q } q 4 # # q 4 # q n q ( q) + q q - 48 Pthgoren tri is: { 4, 48, 50 } 4 { - q, q, + q } hint: Pthgoren tris n e me using ositive integers onl. Forml exlntion: smll integer other smll integer lrgest integer From hint, Pthgoren tris re me using ositive integers. ie. ositive whole numers onl. One of the smller integers is foun using - q If the vlue of ws smller thn the vlue of q, then the nswer woul e negtive. So this oul not e use euse onl ositive whole numers re llowe. Showing using hosen vlues n q : When n q, - q - 3 (this is ositive integer n is llowe) If we sw these roun, so n q - q (this is negtive integer n is not llowe) This will lws hen if the vlue of is smller thn the vlue of q when using Euli s formul. Negtive numers re not llowe euse eh integer reresents the length of the sie of right-ngle tringle. So sie length of -3 oes not mke sense. 6 I SERIES TOPIC Mthletis Pssort 3P Lerning

19 Wht else n ou o? Solutions Pthgors Theorem Pge 8 questions Wheel of Theoorus n so on Mthletis Pssort 3P Lerning I SERIES TOPIC

20 Jigsw Wht else Puzzle n ou o? Solutions Pthgors Theorem Pge 3 questions Squres n right-ngle tringles: Jigsw Puzzle I SERIES TOPIC Mthletis Pssort 3P Lerning

21 Pthgors Theorem Notes Mthletis Pssort 3P Lerning I SERIES TOPIC 9

22 Pthgors Theorem Notes 0 I SERIES TOPIC Mthletis Pssort 3P Lerning

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24 PYTHAGORAS THEOREM APPLICATIONS OF PYTHAGORAS THEOREM.../.../0... APPLICATIONS OF TRIANGLES RIGHT-ANGLED TRIANGLES RIGHT-ANGLED.../.../0... EUCLID S FORMULA FOR PYTHAGOREAN TRAIDS * ", - q, q, + q.../.../0... SQUARES AND RIGHT- ANGLED TRIANGLES SQUARES AND RIGHT- ANGLED TRIANGLES.../.../0... * AWESOME *.../.../0... * AWESOME *