Geometry V1.noteook Ferury 09, 2012 Similr Right Tringles Cn I identify similr tringles in right tringle with the ltitude? Cn I identify the proportions in right tringles? Cn I use the geometri mens theorems to solve prolems? Note the stndrd mrkings of tringle. C If the ltitude is drwn to the hypotenuse of right tringle, then the two tringles formed re similr to the originl tringle nd to eh other. A B If the ltitude is drwn to the hypotenuse of right tringle, then the two tringles formed re similr to the originl tringle nd to eh other. C Tringle KMN ~ Δ ~ Δ M A L N d f e B K 1
Geometry V1.noteook Ferury 09, 2012 Geometri Men Theorem I The length of the ltitude is the geometri men of the lengths of the two segments of the hypotenuse. Geometri Men Theorem I, The length of the ltitude is the geometri men of the lengths of the two segments of the hypotenuse. d f = f e f C A d e B Solve for Geometri Men Theorem II 25 9 The length of eh leg of the right tringle is the geometri men of the lengths of the hypotenuse nd the segment of the hypotenuse tht is djent to the leg. Geometri Men Theorem II, The length of eh leg of the right tringle is the geometri men of the lengths of the hypotenuse nd the segment of the hypotenuse tht is djent to the leg. C Solve for = y = f y A d = d e e = B 64 36 2
Geometry V1.noteook Ferury 09, 2012 Solve for = y = z = 9 y 16 z Cn I identify similr tringles in right tringle with the ltitude? Cn I identify the proportions in right tringles? Cn I use the geometri mens theorems to solve prolems? 3
Geometry V2.noteook Ferury 09, 2012 Cn I prove the Pythgoren Theorem? Cn I use it orretly? The Pythgoren Theorem The Pythgoren Theorem, In right tringle, the squre of the length of the hypotenuse is equl to the sum of the squres of the lengths of the legs. The Pythgoren Theorem, In right tringle, the squre of the length of the hypotenuse is equl to the sum of the squres of the lengths of the legs. 2 + 2 = 2 5 12 13 Given: Any right tringle Prove: 2 + 2 = 2 Sttements Tringle ABC is right A CD perpendiulr to AB = d + e / = /e, / = /d 2 = e, 2 = d 2 + 2 = (d + e) 2 + 2 = 2 Resons Given d f C D e B Perpendiulr line post. Seg. + post. Geometri Men Thm. Cross Multiply + Prop. Su The Grfield Proof President Grfield disovered nother, unique proof to the Pythgoren Theorem. One the set up is omplete, the proof is short nd esy. 1
Geometry V2.noteook Ferury 09, 2012 The Grfield Set Up 1. Mke squre The Grfield Set Up 1. Mke squre 2. Pik onvenient length () nd mrk it off from eh orner (sme diretion) }} }} The Grfield Set Up 1. Mke squre 2. Pik onvenient length () nd mrk it off from eh orner (sme diretion) 3. Mrk the distnes nd the rest. The Grfield Set Up 1. Mke squre 2. Pik onvenient length () nd mrk it off from eh orner (sme diretion) 3. Mrk the distnes nd the rest. 4. Connet the points nd mrk those segments. The Grfield Set Up 1. Mke squre 2. Pik onvenient length () nd mrk it off from eh orner (sme diretion) 3. Mrk the distnes nd the rest. 4. Connet the points nd mrk those segments. Are these four green tringles ongruent to eh other? The Grfield Set Up 1. Mke squre 2. Pik onvenient length () nd mrk it off from eh orner (sme diretion) 3. Mrk the distnes nd the rest. 4. Connet the points nd mrk those segments. Is this qudrilterl squre? Why? Why? 2
Geometry V2.noteook Ferury 09, 2012 The Grfield Proof Using the Pythgoren Theorem orretly! Given: Grfield Set Up Prove: 2 + 2 = 2 Sttements Resons Are = ( + ) 2 Are = 2 + 2 + 2 Are = 4(/2) + 2 2 + 2 + 2 = 2 + 2 2 + 2 = 2 Are of squre Simplify Are is sum of its prts Su. prop. 24 7 X Solve for y: Solve for : Y 6 10 7 5 Solve for z: Cn I prove the Pythgoren Theorem? Cn I use it orretly? 5 z 3
Geometry V3.noteook Ferury 09, 2012 Cn I identify nd generte Pythgoren Triples? Pythgoren Triples The Pythgoren Theorem, In right tringle, the squre of the length of the hypotenuse is equl to the sum of the squres of the lengths of the legs. 5 12 13 Given: Any right tringle Prove: 2 + 2 = 2 Sttements Tringle ABC is right A CD perpendiulr to AB = d + e / = /e, / = /d 2 = e, 2 = d 2 + 2 = (d + e) 2 + 2 = 2 Resons Given d f C D e B Perpendiulr line post. Seg. + post. Geometri Men Thm. Cross Multiply + Prop. Su Pythgoren Triples 2 + 2 = 2 Where positive integers work, they re lled Pythgoren Triples 3 2 + 4 2 = 5 2 6 2 + 8 2 = 10 2 5 2 + 12 2 = 13 2 2 2 + 3 2 = 13 2 not tripple!!! Try these. Do they work? 3 4 5 5 12 13 7 24 25 9 40 41 11 60 61 1
Geometry V3.noteook Ferury 09, 2012 Now, solve for : 3 4 6 8 9 12 9 40 11 60 Now, solve for : 4 5 6 10 8 10 9 15 8 15 Now, solve for : 1 2 2 3 3 5 4 4 8 17 5 13 Here re some tripples. Do you see pttern? 3 4 5 5 12 13 7 24 25 9 40 41 11 60 61 Here re some more tripples. 4 3 5 8 15 17 12 35 37 16 63 65 12 5 13 20 21 29 28 45 53 36 77 85 Now, solve for nd y in these two right tringles: Cn I identify nd generte Pythgoren Triples? y 9 12 20 2
Geometry V4.noteook Ferury 09, 2012 Speil Right Tringles Cn I identify the lengths of the sides of the two speil tringles? Cn I find the re of n equilterl tringle? There re two speil right tringles. This is the first of the two speil right tringles. 45 45 90 Why is it lled 45 45 90? & 30 60 90 Find the length of the hypotenuse: Find the length of the hypotenuse: 2 + 2 = 2 2 4 1
Geometry V4.noteook Ferury 09, 2012 Find the length of the hypotenuse: Find the length of the hypotenuse: 10 n In 45 45 90 Tringle, the hypotenuse is 2 times s long s eh leg. 15 45 o n n ) n ) y ) 8 2 10 z ) 2
Geometry V4.noteook Ferury 09, 2012 Why is this 30 o 60 o 90 o tringle? Find the length of the sides of this 30 o 60 o 90 o tringle: = 4 = = 6 = 10 = = = =?? 6 n 3
Geometry V4.noteook Ferury 09, 2012 n 3 30 o 2n In 30 o 60 o 90 o tringle the hypotenuse is twie s long s the shorter leg nd the longer leg is 3 times s long s the shorter leg. n 30 o 2 8 3 d 5 Are of n equilterl tringle e 12 60 o f S S 3 2 S Are Δ = h 2 S =S(s 3) = S 2 3 2 2 4 4
Geometry V4.noteook Ferury 09, 2012 12 ( 8 ( Are = ( Are = Cn I identify the lengths of the sides of the two speil tringles? Cn I find the re of n equilterl tringle? 5
Geometry V5.noteook Ferury 09, 2012 Gnymede Cllisto Gnymede Cllisto Io Io Trigonometry Trigonometry Cn I define trigonometry? Cn I define ll three trigonometri funtions? Trigonometry is the study of the mesurement of tringles. A trigonometri rtio is the omprison of two sides of right tringle. We will lern three trigonometri funtions. These re funtions of n ngle (ie, enter n ngle nd get numer) 1
Geometry V5.noteook Ferury 09, 2012 They re: sine osine tngent The ngle θ (thet) These re funtions tht del with rtios of right tringle. The hypotenuse θ The djent leg The opposite leg Sine of θ is written s sin θ nd mens The rtio of the Opposite leg to the Hypotenuse For emple θ θ 13 sin θ = 5/13 5 12 60 7 sin θ = 61 11 sin φ = φ (ngle 'phee') Cosine of θ is written s os θ nd mens The rtio of the Adjent leg to the Hypotenuse For emple θ θ 13 os θ = 12/13 5 12 os φ = 60 6 os θ = 61 11 φ 2
Geometry V5.noteook Ferury 09, 2012 Tngent of θ is written s tn θ nd mens The rtio of the Opposite leg to the Adjent leg For emple θ 13 tn θ = 5/12 5 12 tn φ = 60 61 11 φ θ tn θ = 6 The rtio is lwys the sme for given ngle 3 4 10 5 α 15 25 Cn I define trigonometry? Cn I define ll three trigonometri funtions? 8 α 20 α 3
Geometry V6.noteook Sohhto Ferury 09, 2012 Cn I memorize ll three trigonometri funtions? Cn I find ny trig vlue? Cn I solve indiret mesurement prolems using trig? Review the three trig funtions: sine osine tngent These re funtions tht del with rtios of right tringle. The opposite leg The dj e nt l θ eg The hypotenuse Sine of θ is written s sin θ nd mens Cosine of θ is written s os θ nd mens The rtio of the Opposite leg to the Hypotenuse The rtio of the Adjent leg to the Hypotenuse The ngle θ (thet) 1
Geometry V6.noteook Ferury 09, 2012 Tngent of θ is written s tn θ nd mens The rtio of the Opposite leg to the Adjent leg Find these: θ sin θ = 25 φ 24 sin φ = 7 os θ = os φ = tn θ = tn φ = Chief Sohhto SOHCAHTOA Sin = Opposite/Hypotenuse Cos = Adjent/Hypotenuse Tn = Opposite/Adjent Vlues re in you lultors Find these: Ensure tht your ngles re mesured in degrees! (NOT rdins or grdients) Try your lultor: Find sin(56 o ) = os(78 o ) =.2079.8290 α sin α = os α = tn α = 2
Geometry V6.noteook Ferury 09, 2012 Find these: Find these: sin α = sin β = os α = os β = tn α = 25 y tn β = β α Find these: sin β = os β = tn β = Cn I memorize ll three trigonometri funtions? Cn I find ny trig vlue? 16 12 β Cn I solve indiret mesurement prolems using trig? 3
Geometry V7.noteook Ferury 09, 2012 Solving Right Tringles Cn I solve ny right tringle given one ute ngle nd one side? Cn I solve indiret mesurement prolems using trig? Wht does it men to solve tringle? If you know one ngle of right tringle, n you find the other ngle? Find the length of every side nd β the mesure of every ngle! α Wht if you only know two sides? 23 o The Pythgoren Theorem lets you find the third side! 2 + 2 = 2 1
Geometry V7.noteook Ferury 09, 2012 2 + 2 = 2 If you know one ngle nd ny one side, n you find the other two sides? 61 α 60 Use these: sin α = O/H os α = A/H tn α = O/A Solve for : 10 27 o sin α = O/H sin 27 o = /10 10sin 27 o = 10(.4539) = 4.539 = 35 19 o os α = A/H os 19 o = /35 35os 19 o = 35(.9455) = 33.0931 40 51 o 20 45 o tn α = O/A tn 51 o = /20 20tn 51 o = tn α = O/A tn 45 o = 40/ 20(1.2348) = 24.6979 = 40/tn 45 o =? 2
Geometry V7.noteook Ferury 09, 2012 Solve for : 32 o sin α = O/H sin 32 o = 10/ 10/sin 32 o = 10/(.5299) = 18.8707 = 10 25 o 4 os α = A/H os 25 o = 4/ = 4/os 25 o = 4/(.9063) = 4.4135 Use trig for indiret mesurements 1. Set up right tringle 2. Write the est trig funtion 3. Solve Use trig for indiret mesurements 1. Set up right tringle 2. Write the est trig funtion 3. Solve Find the height of the tree. X 30 o 70 feet 40.4145 ft H How tll is this lighthouse? 300 feet 20 o This hot ir lloon is trying to pull silot. When the rope ws fully pulled out, how high ws the lloon over the wter? 102.6060 H 70 o 500 foot rope 171 feet 3
Geometry V7.noteook Ferury 09, 2012 Trig Projet Purpose: To mesure something y indiret mens using trigonometri properties. Requirements: Pik n ojet tht you would like to mesure tht is too lrge (more thn 25 feet) or too diffiult to mesure diretly. Using trig properties, figure wy to get its mesurement. Tke mesurements nd ompute the tul length/height. Find some other wy to verify your mesurements. Write report, showing your work. Give short orl presenttion to the lss (2 4 min). Due Dte: Mrh 22, 2011 (Written report nd orl presenttion) Grding Criteri: Grde will e ssigned s follows. Written Report Cover Pge 10 Desription of the prolem 20 Digrm of the prolem 20 Computtion lerness 20 Computtion ury 20 Orl Report 10 TOTAL POINTS 100 For emple: 5' 28 o 30 feet eye height H Cn I solve ny right tringle given one ute ngle nd one side? Cn I solve indiret mesurement prolems using trig? tn(28 o ) = H/30' = 15.95' H = 15.95 + 5.00' = 20.95' 4
Geometry V8.noteook Ferury 09, 2012 Cn I solve prolems with ngles of depression nd elevtion? Do I know the inverse trig funtions? Angles of Elevtion nd Depresion Yesterdy, we were solving tringles. If you know two sides, you n find the third side. But wht out the ngles? How n we find them? β For most funtions there is n inverse funtion! dd sutrt multiply divide powers roots α If sin θ = A, then sin 1 A = θ We ll this the inverse sine or the rsine If os θ = B, then os 1 B = θ We ll this the inverse osine or the rosine 1
Geometry V8.noteook Ferury 09, 2012 If tn θ = C, then tn 1 C = θ These three new trig funtions re lso in your lultor. Find them! We ll this the inverse tngent or the rtngent These vlues re in you lultors too. Find them How do we use them? Try your lultor: 2 Find sin 1 (.8) = 53.13 o os 1 (.6) = tn 1 (5) = 53.13 o 78.69 o???? α 7 We know: tn α = 2/7 so: tn 1 (2/7) = α 15.94 o Try these: 28 43 12 θ 11 β 40.62 23.55 2
Geometry V8.noteook Angle of elevtion Angle of depression Ferury 09, 2012 With n ngle of elevtion of 65 degrees nd 35 feet of string, how high is the kite? Angle of depression 31.7 feet Angle of elevtion From the top of Ae, it is level to the top of TR. They re 23 feet wy. An ngle of depression of 69o ligns with the ottom of TR. How tll is TR's fe? Cn I solve prolems with ngles of depression nd elevtion? Do I know the inverse trig funtions? The enter of the pyrmid is 1000 m from the oelisk. If the ngle from the top of the pyrmid is 15 degrees. How tll is 268 m the pyrmid? Trig Projet Purpose: To mesure something y indiret mens using trigonometri properties. Requirements: Pik n ojet tht you would like to mesure tht is too lrge (more thn 25 feet) or too diffiult to mesure diretly. Using trig properties, figure wy to get its mesurement. Tke mesurements nd ompute the tul length/height. Find some other wy to verify your mesurements. Write report, showing your work. Give short orl presenttion to the lss. 3
Geometry V8.noteook Ferury 09, 2012 Due Dte: Mrh 22, 2011 (Written report nd orl presenttion) For emple: Grding Criteri: Grde will e ssigned s follows. Written Report Cover Pge 10 Desription of the prolem 20 Digrm of the prolem 20 Computtion lerness 20 Computtion ury 20 Orl Report 10 TOTAL POINTS 100 5' 28 o 30 feet tn(28 o ) = H/30' eye height H = 15.95' H = 15.95 + 5.00' = 20.95' 4
Geometry V9.noteook Ferury 09, 2012 The Converse of the Pythgoren Theorem Cn I determine whether tringle is right, ute, or otuse from its sides? The Pythgoren Theorem Converse, If the squre of the length of the hypotenuse is equl to the sum of the squres of the lengths of the legs, then it is right tringle. Is this right tringle? 28 11? 60 61 21 34 Is this right tringle? If the squre of the length of the longest side of tringle is less thn the sum of the squres of the lengths of the other two sides, then it is n ute tringle. 51 9 30 6 9 8 1
Geometry V9.noteook Ferury 09, 2012 If the squre of the length of the longest side of tringle is less thn the sum of the squres of the lengths of the other two sides, then it is n ute tringle. If the squre of the length of the longest side of tringle is greter thn the sum of the squres of the lengths of the other two sides, then it is n otuse tringle. 6 9 6 11 8? 8 If the squre of the length of the longest side of tringle is greter thn the sum of the squres of the lengths of the other two sides, then it is n otuse tringle. 2 < 2 + 2 2 = 2 + 2 2 > 2 + 2 Aute Tringle Right Tringle Otuse Tringle 6 11 8 Is this tringle ute or otuse? Is this tringle ute or otuse? 5 12 11 10 10 6 2
Geometry V9.noteook Ferury 09, 2012 Try these: Are the tringles right, ute, or otuse? Tringle ABC hs sides: 39, 80, 90 Tringle DEF hs sides: 40, 42, 57 Otuse Aute Cn I determine whether tringle is right, ute, or otuse from its sides? Tringle STU hs sides: 44, 117, 125 Right 3
Geometry V10.noteook Ferury 09, 2012 Io Review of Unit V Review Geometri mens theorems Europ (size of our moon) Gnymede Cllisto (lrger thn Merury) If the ltitude is drwn to the hypotenuse of right tringle, then the two tringles formed re similr to the originl tringle nd to eh other. C Geometri Men Theorem I, The length of the ltitude is the geometri men of the lengths of the two segments of the hypotenuse. d f = f e C A d f e B A d f e B Geometri Men Theorem II, The length of eh leg of the right tringle is the geometri men of the lengths of the hypotenuse nd the segment of the hypotenuse tht is djent to the leg. A d = d f C e e = B Review Geometri mens theorems Pythgoren Theorem The Pythgoren Theorem, In right tringle, the squre of the length of the hypotenuse is equl to the sum of the squres of the lengths of the legs. 2 + 2 = 2 1
Geometry V10.noteook Ferury 09, 2012 Review Geometri mens theorems Pythgoren Theorem Pythgoren Theorem Converse Review Geometri mens theorems Pythgoren Theorem Pythgoren Theorem Converse 2 + 2 > 2 2 + 2 < 2 Review In 45 45 90 Tringle, the hypotenuse Geometri mens theorems Pythgoren Theorem Pythgoren Theorem Converse 2 + 2 > 2 2 + 2 < 2 45 o 45 o 90 o Tringle 30 o 60 o 90 o Tringle n n is 2 times s long s eh leg. ) n ) Review In 30 o 60 o 90 o tringle the hypotenuse is twie s long s the shorter leg nd the longer leg is 3 times s long s the shorter leg. n 3 30 o 2n Geometri mens theorems Pythgoren Theorem Pythgoren Theorem Converse 2 + 2 > 2 2 + 2 < 2 45 o 45 o 90 o Tringle 30 o 60 o 90 o Tringle 6 trig funtions n 2
Geometry V10.noteook Ferury 09, 2012 SOHCAHTOA Sin = Opposite/Hypotenuse Find these: sin β = os β = Cos = Adjent/Hypotenuse tn β = Tn = Opposite/Adjent 7 11 β Review Geometri mens theorems Pythgoren Theorem Pythgoren Theorem Converse 2 + 2 > 2 2 + 2 < 2 45 o 45 o 90 o Tringle 30 o 60 o 90 o Tringle 6 trig funtions Solve right tringles Solve prolems using trig Trig Projet 3