Pthgors Theorem PYTHAGORAS THEOREM www.mthletis.om.u
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
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
How oes it work? Solutions Pthgors Theorem Pge questions Pthgors Theorem for right-ngle tringles 5 9 5 4 0 9 + 8+ 44 5 5 5 4 + 0 96+ 400 5 65 596 Right-ngle Not right-ngle Right-ngle Not right-ngle. 3.4. 3.5 Right-ngle 9.6. + 3.4 50.4+.56 9.6 9.6 6. 9 Not right-ngle Right-ngle 3.. + 3.5.44 +.5 3. 3.69 3. 69 Not right-ngle e f 0 4 9 5 0 + 400 + 44 9 84 84 Right-ngle Not right-ngle 4 + 56 + 49 5 65 65 Right-ngle Not right-ngle Mthletis Pssort 3P Lerning I SERIES TOPIC 3
How oes it work? Solutions Pthgors Theorem Pge 8 questions Pthgors Theorem for right-ngle tringles A J 6 0 0.5 0 K I 4.5 H J 0 + 0. 5 45. 05. 0. 5 + 6 0 400 400 B M K 9 A 5 0 C 5 0 + 5 4 65! 56 L 48 N 9 + 48 8! 304 48 H 5 G 0 48 + 0 5 04 04 The right-ngle tringles re: ΔAJK, ΔHIJ, ΔGHK 4 I SERIES TOPIC Mthletis Pssort 3P Lerning
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 + 6 0 400 400 36 4 5 ΔQRU 5 + 36 39 5 5 P 5 Q 5 6 U ΔRSU ΔSTU 39 + 5 65 45 45 4 + 5 380 380 380 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. + 3 4 5 6 45 mm 05 60 mm 3600 5 mm 565 05 + 3600 565 0 mm 4900 4 mm 56 4 mm 546 4900 + 56 546 5 mm 5 36 mm 96 39 mm 5 5 + 96 5 36 mm 96 mm 599 85 mm 5 96 + 599 5 9 mm 8 40 mm 600 4 mm 68 8 + 600 68 40 mm 600 4 mm 64 58 mm 3364 600 + 64 3364 Mthletis Pssort 3P Lerning I SERIES TOPIC 5
Where oes it work? Solutions Pthgors Theorem Pge questions Clulting the length of the hotenuse 6 + 8 g 8 + 5 36 + 64 g 64 + 5 00 g 89 00 g 89 0 g 5 + 5+ 44. +.6.44 +.56 69 4 69 4 3 Pge questions Clulting the length of the hotenuse 3 h. + 6.0 h h.+ 36 3. h 3. in ext squre root form n + 35 n n 44+ 5 389 n 389 in ext squre root form 4 ( 0 units) + ( 9 units) ( 59. units) + ( 34. units) 00units + 8units 34. 8 units +. 56 units 8units 46. 3 units 8 units 46. 3 units 3. 4536405... units 6809554.... units. 3. 45 units to eiml les. 68. units to eiml les 6 I SERIES TOPIC Mthletis Pssort 3P Lerning
Where oes it work? Solutions Pthgors Theorem Pge 3 questions Clulting the length of the hotenuse 5 Stge Stge Stge 3 40 + 98 39 + 5 36 + 360 600 + 3904 5+ 63504 96 + 9600 40804 6505 30896 40804 6505 30896 0m 55m 369554.... m. 368. m to eiml les The totl length of the 3 stge flight th. 0+ 55+ 36.8m. 88.8 m Pge 5 questions Clulting the length of short sie 6-4 8.5 -.3 66-56.5-.69 00 0.56 00 0. 56 0 84. j 0-56 (8. units) -(8 units) j 4900-336 3.6units -34 units j 64 3.6units j 64 36. units j 4 9. units Mthletis Pssort 3P Lerning I SERIES TOPIC
Where oes it work? Solutions Pthgors Theorem Pge 6 questions Clulting the length of short sie 3 - w 4.5 +.5 89- w 03.065+ 38.065 68 w 65 68 in ext squre root form w 65 in ext squre root form 4 3. 8-83. x 4. 08-3. 4 9044. -6889. x 68. 5664-548. 4968 55. x 390. 55. x 39. 0. 04966... x 33. 50.. 0 to eiml oint x. 33. 8 to eiml oint 8 I SERIES TOPIC Mthletis Pssort 3P Lerning
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 3 4 5 6 5 4. 0 545 544.9 4 4 I M 4. E 3.4 4.9 Q 5 5 53. e 68 3 3 E 6 6. h 0 33 4.6 H 30 g 60 6 Mthletis Pssort 3P Lerning I SERIES TOPIC 9
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 54.030434 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 4.544094... 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 - 4.5 km..35 km 0 I SERIES TOPIC Mthletis Pssort 3P Lerning
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 + 0.89 m B 3.3 m 54.4 m AB AB AB 334.89 m 33489. m 8.3m A 8 m Digrm not rwn to sle. BC (54.4 m) + (3.3 m) BC 959.36 m + 0.89 m BC 90.5m BC 90. 5 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
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 + 4400 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 65-6 34-6 63-30 45-56 56-56 33 3969 900 WY 3969 WX 900 X 65 WY 63 WX 30 34 W 6 I SERIES TOPIC Mthletis Pssort 3P Lerning
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 8.5-6 AD AC -DC. 5-95. BD AD + AB 8 + 6 345. -36 8 m 64 + 36 3065. 00 AC 306. 5 BD 00 AC. 5 m BD 0 m Mthletis Pssort 3P Lerning I SERIES TOPIC 3
Wht else n ou o? Solutions Pthgors Theorem Pge 3 questions Pthgoren tris 0 4 6 0 6 ", 6, 0, 0, 4, 6 sst! Note tht the re written in orer of size. ", 35 3 9 4 ", 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? 49 + 56 65? 65 65 4 + 48 50 96+ 304 500? 500 500? + 34 36? 44 + 56 96? 300! 96 Yes No Yes No Yes No " 5, 36, 39, e " 6, 60, 63, f ", 30, 3, 5 + 36 39? 5+ 96 5? 5 5 6 + 60 63 56+ 3600 3969? 3856! 3969? + 30 3? 44+ 900 96? 044! 96 Yes No Yes No Yes No 4 I SERIES TOPIC Mthletis Pssort 3P Lerning
Wht else n ou o? Solutions Pthgors Theorem Pge 5 questions Euli s formul for Pthgoren tris q - q q + q Tri 3 5 6 3 8-3 # # 4 + 5 { 3, 4, 5 } 3-8 # 3 # 6 3-0 { 6, 8, 0 } 5 - # 5 # 0 5 + 9 { 0,, 9 } - 6 3 # # 6 84 + 6 85 { 3, 84, 85 } - 3 # # 3 66 + 3 30 { 66,, 30 } - 8 # # 8 56 + 8 56 {, 56, 65 } (i) Fin Pthgoren tri in whih n - q is equl to 33 - q 33 - q 33 49 - q 33 49-33 q 6 q q 4 q # # 4 56 + q + 4 65 Pthgoren tri is { 33, 56, 65 } (ii) Fin Pthgoren tri in whih q 5 n + q is equl to 6 - q 6-5 6-5 6 6-5 36 6 q # 6 # 5 60 + q 6-5 Pthgoren tri is {, 60, 6 } Mthletis Pssort 3P Lerning I SERIES TOPIC 5
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 + 50 - 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 - - 3 (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
Wht else n ou o? Solutions Pthgors Theorem Pge 8 questions Wheel of Theoorus 8 6 0 4 n so on Mthletis Pssort 3P Lerning I SERIES TOPIC
Jigsw Wht else Puzzle n ou o? Solutions Pthgors Theorem Pge 3 questions Squres n right-ngle tringles: Jigsw Puzzle 55 4 3 5 3 4 8 I SERIES TOPIC Mthletis Pssort 3P Lerning
Pthgors Theorem Notes Mthletis Pssort 3P Lerning I SERIES TOPIC 9
Pthgors Theorem Notes 0 I SERIES TOPIC Mthletis Pssort 3P Lerning
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 *