8-1. The Pythagorean Theorem and Its Converse. Vocabulary. Review. Vocabulary Builder. Use Your Vocabulary
|
|
- Edwin Fleming
- 7 years ago
- Views:
Transcription
1 8-1 The Pythgoren Theorem nd Its Converse Voulry Review 1. Write the squre nd the positive squre root of eh numer. Numer Squre Positive Squre Root Voulry Builder leg (noun) leg Relted Word: hypotenuse Definition: In right tringle, the sides tht form the right ngle re the legs. Min Ide: The legs of right tringle re perpendiulr. The hypotenuse is the side opposite the right ngle. Use Your Voulry 2. Underline the orret word to omplete the sentene. The hypotenuse is the longest / shortest side in right tringle. Write T for true or F for flse. F T T 3. The hypotenuse of right tringle n e ny one of the three sides. 4. One leg of the tringle t the right hs length 9 m.. The hypotenuse of the tringle t the right hs length 1 m. leg hypotenuse leg 1 m m 9 m Chpter 8 202
2 Theorems 8-1 nd 8-2 Pythgoren Theorem nd Its Converse Pythgoren Theorem If tringle is right tringle, then the sum of the squres of the lengths of the legs is equl to the squre of the length of the hypotenuse. If nabc is right tringle, then Converse of the Pythgoren Theorem If the sum of the squres of the lengths of two sides of tringle is equl to the squre of the length of the third side, then the tringle is right tringle. If , then nabc is right tringle. 6. Cirle the eqution tht shows the orret reltionship mong the lengths of the legs nd the hypotenuse of right tringle Underline the orret words to omplete eh sentene. 7. A tringle with side lengths 3, 4, nd is / is not right tringle euse is equl / not equl to A tringle with side lengths 4,, nd 6 is / is not right tringle euse is equl / not equl to 6 2. A B C Prolem 1 Finding the Length of the Hypotenuse Got It? The legs of right tringle hve lengths 10 nd 24. Wht is the length of the hypotenuse? 9. Lel the tringle t the right. 10. Use the justifitions elow to find the length of the hypotenuse Pythgoren Theorem Sustitute for nd Simplify Add. Tke the positive squre root. 11. The length of the hypotenuse is 26.. One Pythgoren triple is,, nd. If you multiply eh numer y 2, wht numers result? How do the numers tht result ompre to the lengths of the sides of the tringle in Exerises 9 11? 10, 24, 26. Answers my vry. Smple: The numers re the sme s the lengths of the sides of the tringle in Exerises Lesson 8-1
3 Prolem 3 Finding Distne Got It? The size of omputer monitor is the length of its digonl. You wnt to uy 19-in. monitor tht hs height of 11 in. Wht is the width of the monitor? Round to the nerest tenth of n inh. 19 in. 11 in.. Lel the digrm of the omputer monitor t the right. 14. The eqution is solved elow. Write justifition for eh step. in Pythgoren Theorem Sustitute Simplify Sutrt 1 from eh side. Simplify. "240 Tke the positive squre root. < Use lultor. 1. To the nerest tenth of n inh, the width of the monitor is 1. in. Prolem 4 Identifying Right Tringle Got It? A tringle hs side lengths 16, 48, nd 0. Is the tringle right tringle? Explin. 16. Cirle the eqution you will use to determine whether the tringle is right tringle Simplify your eqution from Exerise u Underline the orret words to omplete the sentene. The eqution is true / flse, so the tringle is / is not right tringle. A Pythgoren triple is set of nonzero whole numers,, nd tht stisfy the eqution If you multiply eh numer in Pythgoren triple y the sme whole numer, the three numers tht result lso form Pythgoren triple. Chpter 8 204
4 Theorems 8-3 nd 8-4 Pythgoren Inequlity Theorems Theorem 8-3 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 the tringle is otuse. Theorem 8-4 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 the tringle is ute. Use the figures t the right. Complete eh sentene with ute or otuse. 19. In nabc, , so nabc is In nrst, s 2, r 2 1 t 2, so nrst is 9. otuse ute A R t C S r s T B Lesson Chek Do you UNDERSTAND? Error Anlysis A tringle hs side lengths 16, 34, nd 30. Your friend sys it is not right tringle. Look t your friend s work nd desrie the error. 21. Underline the length tht your friend used s the longest side. Cirle the length of the longest side of the tringle ? = 30 2? = Write the omprison tht your friend should hve used to determine whether the tringle is right tringle Desrie the error in your friend s work. Answers my vry. Smple: My friend used the wrong length for in the omprison. The omprison should e Mth Suess Chek off the voulry words tht you understnd. hypotenuse leg Pythgoren Theorem Pythgoren triple Rte how well you n use the Pythgoren Theorem nd its onverse. Need to review Now I get it! 20 Lesson 8-1
5 8-2 Speil Right Tringles Voulry Review 1. Cirle the segment tht is digonl of squre ABCD. AB AC AD BC CD 2. Underline the orret word to omplete the sentene. A digonl is line segment tht joins two sides / verties of polygon. Voulry Builder D A C B omplement (noun) KAHM pluh munt Other Word Form: omplementry (djetive) Mth Usge: When the mesures of two ngles hve sum of 90, eh ngle is omplement of the other. Nonexmple: Two ngles whose mesures sum to 180 re supplementry. Use Your Voulry Complete eh sttement with the word omplement or omplementry. 3. If m/a 40 nd m/b 0, the ngles re 9. omplementry 4. If m/a 30 nd m/b 60, /B is the 9 of /A. omplement. /P nd /Q re 9 euse the sum of their mesures is 90. omplementry Complete. 6. If /R hs mesure of 3, then the omplement of /R hs mesure of. 7. If /X hs mesure of 22, then the omplement of /X hs mesure of If /C hs mesure of 6, then the omplement of /C hs mesure of Cirle the omplementry ngles Chpter 8 206
6 Theorem Tringle Theorem In tringle, oth legs re ongruent nd the length of the hypotenuse is "2 times the length of leg. s 2 4 s Complete eh sttement for tringle. 10. hypotenuse "2? leg 11. If leg 10, then hypotenuse "2? s Prolem 1 Finding the Length of the Hypotenuse Got It? Wht is the length of the hypotenuse of tringle with leg length!3?. Use the justifitions to find the length of the hypotenuse. hypotenuse "2? leg Tringle Theorem "2? "3 Sustitute. "2? "3 Commuttive Property of Multiplition. "6 Simplify. Prolem 2 Finding the Length of Leg Got It? The length of the hypotenuse of tringle is 10. Wht is the length of one leg?. Will the length of the leg e greter thn or less thn 10? Explin. Less thn. Explntions my vry. Smple: The hypotenuse is the longest side. 14. Use the justifitions to find the length of one leg. hypotenuse "2? leg 10 "2? leg Sustitute Tringle Theorem 10 "2? leg Divide eh side y "2. "2 "2 10 leg "2 Simplify. 10 "2 leg? "2 "2 Multiply y form of 1 to rtionlize the denomintor. 10"2 leg 2 Simplify. leg "2 Divide y Lesson 8-2
7 Prolem 3 Finding Distne Got It? You pln to uild pth long one digonl of 100 ft-y- 100 ft squre grden. To the nerest foot, how long will the pth e? 1. Use the words pth, height, nd width to omplete the digrm. 16. Write L for leg or H for hypotenuse to identify eh prt of the right tringle in the digrm. H pth L height L width 17. Sustitute for hypotenuse nd leg. Let h the length of the hypotenuse. hypotenuse "2? leg h "2? Solve the eqution. Use lultor to find the length of the pth. h!2? 100 h N height width pth 19. To the nerest foot, the length of the pth will e 141 feet. Theorem Tringle Theorem In tringle, the length of the hypotenuse is twie the length of the shorter leg. The length of the longer leg is "3 times the length of the shorter leg. Complete eh sttement for tringle. 20. hypotenuse 2? shorter leg 21. longer leg "3? shorter leg Prolem 4 Think f is the length of the hypotenuse. I n write n eqution relting the hypotenuse nd the shorter leg 3 3 Now I n solve for f. Using the Length of One Side Got It? Wht is the vlue of f in simplest rdil form? 22. Complete the resoning model elow. of the tringle. hypotenuse f f Write 2 2 shorter leg œ s s 30 f s V3 Chpter 8 208
8 Prolem Applying the Tringle Theorem Got It? Jewelry Mking An rtisn mkes pendnts in the shpe of equilterl tringles. Suppose the sides of pendnt re 18 mm long. Wht is the height of the pendnt to the nerest tenth of millimeter? 18 mm 18 mm 23. Cirle the formul you n use to find the height of the pendnt. hypotenuse 2? shorter leg 24. Find the height of the pendnt. longer leg!3? shorter leg 18 mm longer leg "3? shorter leg "3? 9 N To the nerest tenth of millimeter, the height of the pendnt is 1.6 mm. Lesson Chek Do you UNDERSTAND? Resoning A test question sks you to find two side lengths of tringle. You know tht the length of one leg is 6, ut you forgot the speil formul for tringles. Explin how you n still determine the other side lengths. Wht re the other side lengths? 26. Underline the orret word(s) to omplete the sentene. In tringle, the lengths of the legs re different / the sme. 27. Use the Pythgoren Theorem to find the length of the longest side. 28. The other two side lengths re 6 nd 6"2. Mth Suess Chek off the voulry words tht you understnd. leg hypotenuse right tringle Pythgoren Theorem Rte how well you n use the properties of speil right tringles. Need to review longest side: "72 6" Now I get it! 209 Lesson 8-2
9 8-3 Trigonometry Voulry Review The Venn digrm t the right shows the reltionship etween similr nd ongruent figures. Write T for true or F for flse. F T 1. All similr figures re ongruent figures. 2. All ongruent figures re similr figures. Similr Figures Congruent Figures T 3. Some similr figures re ongruent figures. 4. Cirle the postulte or theorem you n use to verify tht the tringles t the right re similr. AA, Postulte SAS, Theorem SSS, Theorem Voulry Builder rtio (noun) RAY shee oh Relted Words: rte, rtionl Definition: A rtio is the omprison of two quntities y division. Exmple: If there re 6 tringles nd squres, the rtio of tringles to squres is 6 nd the rtio of squres to tringles is 6. Use Your Voulry Use the tringle t the right for Exerises 7.. Cirle the rtio of the length of the longer leg to the length of the shorter leg. 6. Cirle the rtio of the length of the shorter leg to the length of the hypotenuse. 7. Cirle the rtio of the length of the longer leg to the length of the hypotenuse. Chpter 8 210
10 Key Conept The Trigonometri Rtios sine of /A osine of /A tngent of /A length of leg opposite/a length of hypotenuse length of leg djent to/a length of hypotenuse length of leg opposite/a length of leg djent to/a A B C Drw line from eh trigonometri rtio in Column A to its orresponding rtio in Column B. Column A 8. sin B 9. os B 10. tn B Column B 11. Resoning Suppose nabc is right isoseles tringle. Wht would the tngent of /B equl? Explin. Explntions my vry. Smple: 1. The legs would e ongruent, so would equl 1. Prolem 1 Writing Trigonometri Rtios Got It? Wht re the sine, osine, nd tngent rtios for lg?. Cirle the mesure of the leg opposite /G Cirle the mesure of the hypotenuse Cirle the mesure of the leg djent to /G Write eh trigonometri rtio. sin G os G opposite hypotenuse djent hypotenuse tn G opposite djent T 17 1 G 8 R 211 Lesson 8-3
11 Prolem 2 Using Trigonometri Rtio to Find Distne Got It? Find the vlue of w to the nerest tenth. Below is one student s solution w os 4 w 17 os 4 (17) w w 10 w 16. Cirle the trigonometri rtio tht uses sides w nd 17. sin 48 os 48 tn Wht error did the student mke? Answers my vry. Smple: The student wrote os 4 w 17 rther thn sin 4 w Find the vlue of w orretly. sin 4 w 17 sin 4 (17) w N w.8 N w 19. The vlue of w to the nerest tenth is.8. Prolem 3 Using Inverses Got It? Use the figure elow. Wht is mly to the nerest degree? P 100 T 41 Y 20. Cirle the lengths tht you know. hypotenuse side djent to /Y side opposite /Y 21. Cross out the rtios tht you will NOT use to find m/y. sine osine tngent 22. Underline the orret word to omplete the sttement. If you know the sine, osine, or tngent rtio of n ngle, you n use the inverse / rtio to find the mesure of the ngle. Chpter 8 2
12 23. Follow the steps to find m/y. 1 Write the rtio. 100 tn Y 41 2 Use the inverse. 100 Y tn ( 1 41 ) 3 Use lultor. Y To the nerest degree, m/y < 68. Lesson Chek Do you UNDERSTAND? Error Anlysis A student sttes tht sin A S sin X euse the lengths of the sides of kabc re greter thn the lengths of the sides of kxyz. Wht is the student s error? Explin. Y B Underline the orret word(s) to omplete eh sentene. 2. nabc nd nxyz re / re not similr. Z 3 X C 3 A 26. /A nd /X re / re not ongruent, so sin 38 is / is not equl to sin Wht is the student s error? Explin. Answers my vry. Smple: The student did not look t the mesures of la nd lx. Congruent ngles hve equl sine rtios. Mth Suess Chek off the voulry words tht you understnd. trigonometri rtios sine osine tngent Rte how well you n use trigonometri rtios. Need to review Now I get it! 2 Lesson 8-3
13 8-4 Angles of Elevtion nd Depression Voulry Review Underline the orret word(s) or numer to omplete eh sentene. 1. The mesure of right ngle is greter / less thn the mesure of n ute ngle nd greter / less thn the mesure of n otuse ngle. 2. A right ngle hs mesure of 4 / 90 / Lines tht interset to form four right ngles re prllel / perpendiulr lines. 4. Cirle the right ngle(s) in the figure. /ACB /ADB /BAC A /BAD /CBA /DBA Voulry Builder D B C elevtion (noun) el uh VAY shun Relted Word: depression Definition: The elevtion of n ojet is its height ove given level, suh s eye level or se level. Mth Usge: Angles of elevtion nd depression re ute ngles of right tringles formed y horizontl distne nd vertil height. Use Your Voulry Complete eh sttement with the orret word from the list elow. Use eh word only one. elevte elevted elevtion. John 9 his feet on footstool. 6. The 9 of Mt MKinley is 20,320 ft. 7. You 9 n ojet y rising it to higher position. elevted elevtion elevte Chpter 8 214
14 Prolem 1 Identifying Angles of Elevtion nd Depression Got It? Wht is desription of l2 s it reltes to the sitution shown? Write T for true or F for flse. T 8. /2 is ove the horizontl line. T F 9. /2 is the ngle of elevtion from the person in the hot-ir lloon to the ird. 10. /2 is the ngle of depression from the person in the hot-ir lloon to the ird. F 11. /2 is the ngle of elevtion from the top of the mountin to the person in the hot-ir lloon.. Desrie /2 s it reltes to the sitution shown. Answers my vry. Smple: l2 is the ngle of elevtion from the person in the hot-ir lloon to the ird. Prolem 2 Using the Angle of Elevtion Got It? You sight rok limer on liff t 32 ngle of elevtion. Your eye level is 6 ft ove the ground nd you re 1000 feet from the se of the liff. Wht is the pproximte height of the rok limer from the ground?. Use the informtion in the prolem to omplete the prolem-solving model elow. Eye level ft Know Need Pln Angle of elevtion Height of limer from Find the length of the is the ground leg opposite 328 y using tn Distne to the liff is 1000 ft. Then dd 6 ft. Eye level is 6 ove the ground. ft Climer 21 Lesson 8-4
15 14. Explin why you use tn 328 nd not sin 328 or os 328. Answers my vry. Smple: The sine rtio involves two unknowns. The osine rtio involves the hypotenuse nd 1000, ut I do not wnt to know the hypotenuse. The rtio tht uses the unknown height nd 1000 is the tngent rtio. 1. The prolem is solved elow. Use one of the resons from the list t the right to justify eh step. tn 328 d 1000 Write the eqution. Solve for d. Use lultor. Write the eqution. (tn 328) 1000 d Solve for d. d < Use lultor. 16. The height from your eye level to the limer is out 62 ft. 17. The height of the rok limer from the ground is out 631 ft. Prolem 3 Using the Angle of Depression Got It? An irplne pilot sights life rft t 26 ngle of depression. The irplne s ltitude is 3 km. Wht is the irplne s horizontl distne d from the rft? 18. Lel the digrm elow. ltitude 3 km Not to sle 26º Angle of elevtion 26º d Angle of depression horizontl distne Rft 19. Cirle the eqution you ould use to find the horizontl distne d. sin d 20. Solve your eqution from Exerise 19. tn d d 3 tn 268 d os d tn d 21. To the nerest tenth, the irplne s horizontl distne from the rft is 6.2 km. Chpter 8 216
16 Lesson Chek Do you UNDERSTAND? Voulry How is n ngle of elevtion formed? Underline the orret word(s) to omplete eh sentene. 22. The ngle of elevtion is formed ove / elow horizontl line. 23. The ngle of depression is formed ove / elow horizontl line. 24. The mesure of n ngle of elevtion is equl to / greter thn / less thn the mesure of the ngle of depression. Lesson Chek Do you UNDERSTAND? Error Anlysis A homework question sys tht the ngle of depression from the ottom of house window to ll on the ground is 20. At the right is your friend s sketh of the sitution. Desrie your friend s error. 2. Is the ngle tht your friend identified s the ngle of depression formed y the horizontl nd the line of sight? Yes / No 26. Is the orret ngle of depression djent to or opposite the ngle identified y your friend? 27. Desrie your friend s error. Mth Suess Chek off the voulry words tht you understnd. ngle of elevtion ngle of depression trigonometri rtios Rte how well you n use ngles of elevtion nd depression. Need to review Now I get it! 20 djent to / opposite Answers my vry. Smple: My friend identified the wrong ngle. The orret ngle of depression is elow the horizontl line. 217 Lesson 8-4
17 Lw of Sines 8- Voulry Review 1. Drw line segment from eh ngle of the tringle to its opposite side. C A B 2. Cirle the orret word. A rtio is the omprison of two quntities y ddition sutrtion multiplition division sine (noun) syn Relted Words: tringle, side length, ngle mesure, opposite, osine Definition: In right tringle, sine is the rtio of the side opposite given ute ngle to the hypotenuse. Exmple: If you know the mesure of n ute ngle of right tringle nd the length of the opposite side, you n use the sine rtio to find the length of the hypotenuse. Use Your Voulry 3. A tringle hs given ute ngle. Cirle its sine rtio. djent hypotenuse hypotenuse opposite opposite hypotenuse opposite djent 4. A right tringle hs one ute ngle mesuring The length of the side djent to this ngle is 4 units, nd the length of the side opposite this ngle is 3 units. The length of the hypotenuse is units. Cirle the sine rtio of the ngle. 43 Chpter Voulry Builder
18 Lw of Sines For ny nabc, let the lengths of the sides opposite ngles A, B, nd C e,, nd, respetively. Then the Lw of Sines reltes the sine of eh ngle to the length of its opposite side. sin A sin B sin C. If you know 2 ngles nd 1 side of tringle, n you find ll of the missing mesures? Explin. Yes; sine the sum of tringle s ngles re 1808, you n find the third ngle; then use the Lw of Sines to find the other 2 sides. A C B Prolem 1 Using the Lw of Sines (AAS) Got It? In DABC, m/a 48, m/b 93, nd AC 1. Wht is AB to the nerest tenth? 6. Find nd lel m/c C 7. Lel side lengths,, nd. Whih side is the length of AB? 8. Cirle the eqution whih n e used to solve this prolem. Explin your resoning. sin C sin A sin C sin B sin B sin A B 1 48 A Answers my vry. Smple: I wnt to pik the eqution in whih I hve 3 out of the 4 vlues. 9. Reple the vriles in the eqution with vlues from DABC. Prolem 2 sin 398 sin Find the sine vlues of the given ngles, ross multiply, then solve for. < (.6293 ) ( 1 ) < (.9986 ) 11. The length of AB is out 9. units Using the Lw of Sines (SSA) Got It? In DKLM, LM 9, KM 14, nd m/l 10. To the nerest tenth, wht is m/k? m L 10 9 k. Lel the tringle with informtion from the prolem nd the length of the sides s k, l, m. K 14 l M 219 Lesson 8-
19 . Use the letter tht represents the length of KM to write pir of rtios using some of the letters k, l, m, K, L nd M. sin L sin K l k 14. Fill in the vlues in the eqution from Exerise nd solve for sin K. sin sin K 9 sin K < (.969)(9) 14 <.6210 sin K < Use your lultor nd tke the inverse sine of oth sides of the eqution to find m/k. sin 2 1 (sin K) < sin , therefore m/k < 38.4 Prolem 3 Using the Lw of Sines to Solve Prolem Got It? The right-fielder fields softll etween first se nd seond se s shown in the figure. If the right-fielder throws the ll to seond se, how fr does she throw the ll? 16. Underline the orret word to omplete eh sentene. In this prolem, the solution is side / ngle. To find the solution, I need to first find missing side / ngle. 17. In order to use the Lw of Sines wht informtion will you need tht is missing nd why? I need to find the mesure of the ngle formed y seond se, the right fielder, nd first se euse I need the mesure of the ngle opposite the given side to use the Lw of Sines. 18. Cirle the eqution you ould use to solve for the missing solution. sin sin 408 sin sin 688 sin sin Fill in the lnks to omplete the eqution. Then solve the eqution nd find the solution. sin 728 sin nd Bse (0.911) < 60(0.6428) < ft Right-fielder st Bse Kimmy throws the ll out 40.6 feet. Chpter 8 220
20 Lesson Chek Do you UNDERSTAND? Resoning If you know the three side lengths of tringle, n you use the Lw of Sines to find the missing ngle mesures? Explin. 20. Wht do AAS, ASA, nd SSA stnd for? Mth eh term with its definition. Then tell wht the three terms hve in ommon. AAS Side-Side-Angle ASA Angle-Angle-Side SSA Angle-Side-Angle They ll inlude sides nd ngles. 21. If you know only the three side lengths of tringle, n you use the Lw of Sines to find the missing ngle mesures? Explin. Answers my vry. Smple: No; To use the Lw of Sines, you need 2 ngles nd 1 side or 2 sides nd 1 ngle. Error Anlysis In DPQR, PQ 4 m, QR 3 m, nd m/r 7. Your friend uses the Lw of Sines to write Explin the error. sin 78 sin x8 4 to find m/q. 3 P 22. Lel the digrm with the given informtion. Did your friend orretly mth the ngles nd the sides? 4 Answers my vry. Smple: The side length opposite /R is 4, _ not 3. /Q is not opposide side length 4. _ 7 R x 3 Q Mth Suess Chek off the voulry words tht you understnd. Lw of Sines rtio djent inverse sine Rte how well you n use the Lw of Sines. Need to review Now I get it! Lesson 8-
21 Lw of Cosines 8-6 Voulry Review Look t DABC. C A B nd 1. Nme the sides tht re djent to ngle A. 2. Whih side is opposite of ngle B? 3. Identify eh ngle mesure s ute, right, or otuse. 48 ute 1008 otuse 908 right Cosine (noun) KOH syn Relted Word: tringle, side length, ngle mesure, opposite, sine Definition: In right tringle, osine is the rtio of the side djent to given ute ngle to the hypotenuse. Exmple: If you know the mesure of n ute ngle of right tringle nd the length of the djent side, you n use the osine rtio to find the length of the hypotenuse. Use Your Voulry 4. A tringle hs given ute ngle. Cirle its osine rtio. djent hypotenuse hypotenuse djent opposite hypotenuse djent opposite. A right tringle hs one ute ngle mesuring 3.18, the length of the side djent to this ngle is 9 units, nd the length of the side opposite this ngle is units. The length of the hypotenuse is 1 units. Cirle the osine rtio of the 3.18 ngle. 9 Chpter Voulry Builder
22 Lw of Cosines For ny nabc with side lengths,, nd opposite ngles A, B, nd C, respetively, the Lw of Cosines reltes the mesures of the tringles ording to the following equtions os A C os B os C A 6. Cirle the eqution tht is true for DDEF. d 2 f 2 1 e 2 2 2de os D E f 2 d 2 1 e 2 2 2de os F d f e 2 d 2 1 f 2 2 df os E F e B D Prolem 1 Using the Lw of Cosines (SAS) Got It? In DLMN, m/l 1048, LM 48, nd LN 29. Find MN to the nerest tenth. 7. Lel the sides of DLMN with the letters l, m, nd n. 8. Use the informtion in the prolem to omplete the prolem-solving model elow. M n 48 l 104 L 29 N m Know LM is opposite / LM 48 letter LN is opposite / LN 29 letter 9. Find MN y solving for l.. I 2 m 2 1 n 2 2 2( m )( n ) os /L. Write n eqution using l, m, n, nd /L.. l ( 29 )( 48 ) os 104. Sustitute the vlues from the tringle.. l 2 < (2784? ). Use the Order of Opertions nd l 2 l 2 < N M n m < Need MN letter An eqution using letters l, m, n, nd /L. I solve for l 2. Pln Beuse you know m/ L nd need MN, sustitute the ngle mesure nd the two side lengths into the eqution nd solve for l. d. l < 61.8 MN d. Tke the squre root of oth sides. 223 Lesson 8-6
23 Prolem 2 Using the Lw of Cosines (SSS) Got It? In DTUV ove, find m/t to the nerest tenth of degree. 10. Lel the sides of the tringle with t, u, nd r. 11. Solve for m/t following the given STEPS. t 2 u 2 1 v 2 2 2( u )( v ) os T Write n eqution using the Lw of Cosines ( 6.7 )( 4.4 ) os T Sustitute the vlues from the tringle os T Simplify y squring nd multiplying os T Add the first two numers. U 4.4 v T t u V os T Get oeffiient of os T nd os T lone os T Divide y the oeffiient of os T. os < T Tke the inverse osine of oth sides of the eqution. m/t < 76.4 Prolem 3 Using the Lw of Cosines to Solve Prolem Got It? You nd friend hike 1.4 miles due west from mpsite. At the sme time two other friends hike 1.9 miles t heding of S 118W (118west of south) from the mpsite. To the nerest tenth of mile, how fr prt re the two groups?. Lel the model with informtion from the prolem nd letter the ngles nd sides. West = 1.4 miles = 1.9 miles 11 mpsite South. Find the mesure of the ngle tht is the omplement of the 118ngle Chpter 8 224
24 14. Write nd solve n eqution for finding the distne etween the two groups (1.4)(1.9) os(79) (.191) < 2. miles Lesson Chek Do you UNDERSTAND? Writing Explin how you hoose etween the Lw of Sines nd the Lw of Cosines when finding the mesure of missing ngle or side. 1. Write C if you would use the Lw of Cosines to find missing mesure in tringle or S if you would use the Lw of Sines. C The lengths of two sides nd the mesure of the inluded ngle re given. Find the length of the third side. C S The lengths of three sides re given. Find the mesure of one ngle. The mesures of two ngles nd the length of the inluded side re given. Find the length of nother side. 16. Explin how to hoose etween the Lw of Sines nd the Lw of Cosines in solving tringle. Answers my vry. Smple: Use the Lw of Cosines when you know SAS _ or SSS. Use the Lw of Sines when you know SSA or ASA. _ Mth Suess Chek off the voulry words tht you understnd. Lw of Cosines Lw of Sines trigonometry Rte how well you n use the Lw of Cosines. Need to review Now I get it! Lesson 8-6
Geometry 7-1 Geometric Mean and the Pythagorean Theorem
Geometry 7-1 Geometric Men nd the Pythgoren Theorem. Geometric Men 1. Def: The geometric men etween two positive numers nd is the positive numer x where: = x. x Ex 1: Find the geometric men etween the
More informationThe remaining two sides of the right triangle are called the legs of the right triangle.
10 MODULE 6. RADICAL EXPRESSIONS 6 Pythgoren Theorem The Pythgoren Theorem An ngle tht mesures 90 degrees is lled right ngle. If one of the ngles of tringle is right ngle, then the tringle is lled right
More informationSECTION 7-2 Law of Cosines
516 7 Additionl Topis in Trigonometry h d sin s () tn h h d 50. Surveying. The lyout in the figure t right is used to determine n inessile height h when seline d in plne perpendiulr to h n e estlished
More information1. Definition, Basic concepts, Types 2. Addition and Subtraction of Matrices 3. Scalar Multiplication 4. Assignment and answer key 5.
. Definition, Bsi onepts, Types. Addition nd Sutrtion of Mtries. Slr Multiplition. Assignment nd nswer key. Mtrix Multiplition. Assignment nd nswer key. Determinnt x x (digonl, minors, properties) summry
More informationLesson 2.1 Inductive Reasoning
Lesson.1 Inutive Resoning Nme Perio Dte For Eerises 1 7, use inutive resoning to fin the net two terms in eh sequene. 1. 4, 8, 1, 16,,. 400, 00, 100, 0,,,. 1 8, 7, 1, 4,, 4.,,, 1, 1, 0,,. 60, 180, 10,
More informationAngles 2.1. Exercise 2.1... Find the size of the lettered angles. Give reasons for your answers. a) b) c) Example
2.1 Angles Reognise lternte n orresponing ngles Key wors prllel lternte orresponing vertilly opposite Rememer, prllel lines re stright lines whih never meet or ross. The rrows show tht the lines re prllel
More informationAngles and Triangles
nges nd Tringes n nge is formed when two rys hve ommon strting point or vertex. The mesure of n nge is given in degrees, with ompete revoution representing 360 degrees. Some fmiir nges inude nother fmiir
More informationSOLVING EQUATIONS BY FACTORING
316 (5-60) Chpter 5 Exponents nd Polynomils 5.9 SOLVING EQUATIONS BY FACTORING In this setion The Zero Ftor Property Applitions helpful hint Note tht the zero ftor property is our seond exmple of getting
More informationExample 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers.
2 Rtionl Numbers Integers such s 5 were importnt when solving the eqution x+5 = 0. In similr wy, frctions re importnt for solving equtions like 2x = 1. Wht bout equtions like 2x + 1 = 0? Equtions of this
More informationChapter. Contents: A Constructing decimal numbers
Chpter 9 Deimls Contents: A Construting deiml numers B Representing deiml numers C Deiml urreny D Using numer line E Ordering deimls F Rounding deiml numers G Converting deimls to frtions H Converting
More informationWords Symbols Diagram. abcde. a + b + c + d + e
Logi Gtes nd Properties We will e using logil opertions to uild mhines tht n do rithmeti lultions. It s useful to think of these opertions s si omponents tht n e hooked together into omplex networks. To
More information. At first sight a! b seems an unwieldy formula but use of the following mnemonic will possibly help. a 1 a 2 a 3 a 1 a 2
7 CHAPTER THREE. Cross Product Given two vectors = (,, nd = (,, in R, the cross product of nd written! is defined to e: " = (!,!,! Note! clled cross is VECTOR (unlike which is sclr. Exmple (,, " (4,5,6
More informationPractice Test 2. a. 12 kn b. 17 kn c. 13 kn d. 5.0 kn e. 49 kn
Prtie Test 2 1. A highwy urve hs rdius of 0.14 km nd is unnked. A r weighing 12 kn goes round the urve t speed of 24 m/s without slipping. Wht is the mgnitude of the horizontl fore of the rod on the r?
More informationRatio and Proportion
Rtio nd Proportion Rtio: The onept of rtio ours frequently nd in wide vriety of wys For exmple: A newspper reports tht the rtio of Repulins to Demorts on ertin Congressionl ommittee is 3 to The student/fulty
More informationMathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100
hsn.uk.net Higher Mthemtics UNIT 3 OUTCOME 1 Vectors Contents Vectors 18 1 Vectors nd Sclrs 18 Components 18 3 Mgnitude 130 4 Equl Vectors 131 5 Addition nd Subtrction of Vectors 13 6 Multipliction by
More informationMath 314, Homework Assignment 1. 1. Prove that two nonvertical lines are perpendicular if and only if the product of their slopes is 1.
Mth 4, Homework Assignment. Prove tht two nonverticl lines re perpendiculr if nd only if the product of their slopes is. Proof. Let l nd l e nonverticl lines in R of slopes m nd m, respectively. Suppose
More informationReasoning to Solve Equations and Inequalities
Lesson4 Resoning to Solve Equtions nd Inequlities In erlier work in this unit, you modeled situtions with severl vriles nd equtions. For exmple, suppose you were given usiness plns for concert showing
More informationInterior and exterior angles add up to 180. Level 5 exterior angle
22 ngles n proof Ientify interior n exterior ngles in tringles n qurilterls lulte interior n exterior ngles of tringles n qurilterls Unerstn the ie of proof Reognise the ifferene etween onventions, efinitions
More informationUse Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions.
Lerning Objectives Loci nd Conics Lesson 3: The Ellipse Level: Preclculus Time required: 120 minutes In this lesson, students will generlize their knowledge of the circle to the ellipse. The prmetric nd
More informationLesson 4.1 Triangle Sum Conjecture
Lesson 4.1 ringle um onjecture Nme eriod te n ercises 1 9, determine the ngle mesures. 1. p, q 2., y 3., b 31 82 p 98 q 28 53 y 17 79 23 50 b 4. r, s, 5., y 6. y t t s r 100 85 100 y 30 4 7 y 31 7. s 8.
More informationSection 5-4 Trigonometric Functions
5- Trigonometric Functions Section 5- Trigonometric Functions Definition of the Trigonometric Functions Clcultor Evlution of Trigonometric Functions Definition of the Trigonometric Functions Alternte Form
More informationRIGHT TRIANGLES AND THE PYTHAGOREAN TRIPLETS
RIGHT TRIANGLES AND THE PYTHAGOREAN TRIPLETS Known for over 500 yers is the fct tht the sum of the squres of the legs of right tringle equls the squre of the hypotenuse. Tht is +b c. A simple proof is
More informationPROBLEMS 13 - APPLICATIONS OF DERIVATIVES Page 1
PROBLEMS - APPLICATIONS OF DERIVATIVES Pge ( ) Wter seeps out of conicl filter t the constnt rte of 5 cc / sec. When the height of wter level in the cone is 5 cm, find the rte t which the height decreses.
More informationSOLVING QUADRATIC EQUATIONS BY FACTORING
6.6 Solving Qudrti Equtions y Ftoring (6 31) 307 In this setion The Zero Ftor Property Applitions 6.6 SOLVING QUADRATIC EQUATIONS BY FACTORING The tehniques of ftoring n e used to solve equtions involving
More informationOperations with Polynomials
38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: Write polynomils in stndrd form nd identify the leding coefficients nd degrees of polynomils Add nd subtrct polynomils Multiply
More informationMATH PLACEMENT REVIEW GUIDE
MATH PLACEMENT REVIEW GUIDE This guie is intene s fous for your review efore tking the plement test. The questions presente here my not e on the plement test. Although si skills lultor is provie for your
More informationEnd of term: TEST A. Year 4. Name Class Date. Complete the missing numbers in the sequences below.
End of term: TEST A You will need penil nd ruler. Yer Nme Clss Dte Complete the missing numers in the sequenes elow. 8 30 3 28 2 9 25 00 75 25 2 Put irle round ll of the following shpes whih hve 3 shded.
More informationThe Pythagorean Theorem
The Pythgoren Theorem Pythgors ws Greek mthemtiin nd philosopher, orn on the islnd of Smos (. 58 BC). He founded numer of shools, one in prtiulr in town in southern Itly lled Crotone, whose memers eventully
More informationLaw of Cosines. If the included angle is a right angle then the Law of Cosines is the same as the Pythagorean Theorem.
Law of Cosines In the previous section, we learned how the Law of Sines could be used to solve oblique triangles in three different situations () where a side and two angles (SAA) were known, () where
More informationCS99S Laboratory 2 Preparation Copyright W. J. Dally 2001 October 1, 2001
CS99S Lortory 2 Preprtion Copyright W. J. Dlly 2 Octoer, 2 Ojectives:. Understnd the principle of sttic CMOS gte circuits 2. Build simple logic gtes from MOS trnsistors 3. Evlute these gtes to oserve logic
More informationHow To Find The Re Of Tringle
Heron s Formul for Tringulr Are y Christy Willims, Crystl Holom, nd Kyl Gifford Heron of Alexndri Physiist, mthemtiin, nd engineer Tught t the museum in Alexndri Interests were more prtil (mehnis, engineering,
More informationEQUATIONS OF LINES AND PLANES
EQUATIONS OF LINES AND PLANES MATH 195, SECTION 59 (VIPUL NAIK) Corresponding mteril in the ook: Section 12.5. Wht students should definitely get: Prmetric eqution of line given in point-direction nd twopoint
More informationRadius of the Earth - Radii Used in Geodesy James R. Clynch Naval Postgraduate School, 2002
dius of the Erth - dii Used in Geodesy Jmes. Clynh vl Postgrdute Shool, 00 I. Three dii of Erth nd Their Use There re three rdii tht ome into use in geodesy. These re funtion of ltitude in the ellipsoidl
More information1. Find the zeros Find roots. Set function = 0, factor or use quadratic equation if quadratic, graph to find zeros on calculator
AP Clculus Finl Review Sheet When you see the words. This is wht you think of doing. Find the zeros Find roots. Set function =, fctor or use qudrtic eqution if qudrtic, grph to find zeros on clcultor.
More informationHomework 3 Solutions
CS 341: Foundtions of Computer Science II Prof. Mrvin Nkym Homework 3 Solutions 1. Give NFAs with the specified numer of sttes recognizing ech of the following lnguges. In ll cses, the lphet is Σ = {,1}.
More informationCalculating Principal Strains using a Rectangular Strain Gage Rosette
Clulting Prinipl Strins using Retngulr Strin Gge Rosette Strin gge rosettes re used often in engineering prtie to determine strin sttes t speifi points on struture. Figure illustrtes three ommonly used
More information1 Fractions from an advanced point of view
1 Frtions from n vne point of view We re going to stuy frtions from the viewpoint of moern lger, or strt lger. Our gol is to evelop eeper unerstning of wht n men. One onsequene of our eeper unerstning
More informationIf two triangles are perspective from a point, then they are also perspective from a line.
Mth 487 hter 4 Prtie Prolem Solutions 1. Give the definition of eh of the following terms: () omlete qudrngle omlete qudrngle is set of four oints, no three of whih re olliner, nd the six lines inident
More informationVectors Summary. Projection vector AC = ( Shortest distance from B to line A C D [OR = where m1. and m
. Slr prout (ot prout): = osθ Vetors Summry Lws of ot prout: (i) = (ii) ( ) = = (iii) = (ngle etween two ientil vetors is egrees) (iv) = n re perpeniulr Applitions: (i) Projetion vetor: B Length of projetion
More informationAppendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered:
Appendi D: Completing the Squre nd the Qudrtic Formul Fctoring qudrtic epressions such s: + 6 + 8 ws one of the topics introduced in Appendi C. Fctoring qudrtic epressions is useful skill tht cn help you
More informationUnit 6: Exponents and Radicals
Eponents nd Rdicls -: The Rel Numer Sstem Unit : Eponents nd Rdicls Pure Mth 0 Notes Nturl Numers (N): - counting numers. {,,,,, } Whole Numers (W): - counting numers with 0. {0,,,,,, } Integers (I): -
More informationLINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES
LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES DAVID WEBB CONTENTS Liner trnsformtions 2 The representing mtrix of liner trnsformtion 3 3 An ppliction: reflections in the plne 6 4 The lgebr of
More informationMaximum area of polygon
Mimum re of polygon Suppose I give you n stiks. They might e of ifferent lengths, or the sme length, or some the sme s others, et. Now there re lots of polygons you n form with those stiks. Your jo is
More informationPhysics 43 Homework Set 9 Chapter 40 Key
Physics 43 Homework Set 9 Chpter 4 Key. The wve function for n electron tht is confined to x nm is. Find the normliztion constnt. b. Wht is the probbility of finding the electron in. nm-wide region t x
More informationExperiment 6: Friction
Experiment 6: Friction In previous lbs we studied Newton s lws in n idel setting, tht is, one where friction nd ir resistnce were ignored. However, from our everydy experience with motion, we know tht
More informationVectors 2. 1. Recap of vectors
Vectors 2. Recp of vectors Vectors re directed line segments - they cn be represented in component form or by direction nd mgnitude. We cn use trigonometry nd Pythgors theorem to switch between the forms
More informationPolynomial Functions. Polynomial functions in one variable can be written in expanded form as ( )
Polynomil Functions Polynomil functions in one vrible cn be written in expnded form s n n 1 n 2 2 f x = x + x + x + + x + x+ n n 1 n 2 2 1 0 Exmples of polynomils in expnded form re nd 3 8 7 4 = 5 4 +
More informationMultiplication and Division - Left to Right. Addition and Subtraction - Left to Right.
Order of Opertions r of Opertions Alger P lese Prenthesis - Do ll grouped opertions first. E cuse Eponents - Second M D er Multipliction nd Division - Left to Right. A unt S hniqu Addition nd Sutrction
More informationBinary Representation of Numbers Autar Kaw
Binry Representtion of Numbers Autr Kw After reding this chpter, you should be ble to: 1. convert bse- rel number to its binry representtion,. convert binry number to n equivlent bse- number. In everydy
More information11.2 The Law of Sines
894 Applitions of Trigonometry 11. The Lw of Sines Trigonometry literlly mens mesuring tringles nd with Chpter 10 under our belts, we re more thn prepred to do just tht. The min gol of this setion nd the
More informationTRIGONOMETRY OF THE RIGHT TRIANGLE
HPTER 8 HPTER TLE OF ONTENTS 8-1 The Pythagorean Theorem 8-2 The Tangent Ratio 8-3 pplications of the Tangent Ratio 8-4 The Sine and osine Ratios 8-5 pplications of the Sine and osine Ratios 8-6 Solving
More informationWarm-up for Differential Calculus
Summer Assignment Wrm-up for Differentil Clculus Who should complete this pcket? Students who hve completed Functions or Honors Functions nd will be tking Differentil Clculus in the fll of 015. Due Dte:
More informationONLINE PAGE PROOFS. Trigonometry. 6.1 Overview. topic 6. Why learn this? What do you know? Learning sequence. measurement and geometry
mesurement nd geometry topic 6 Trigonometry 6.1 Overview Why lern this? Pythgors ws gret mthemticin nd philosopher who lived in the 6th century BCE. He is est known for the theorem tht ers his nme. It
More informationPROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY
MAT 0630 INTERNET RESOURCES, REVIEW OF CONCEPTS AND COMMON MISTAKES PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY Contents 1. ACT Compss Prctice Tests 1 2. Common Mistkes 2 3. Distributive
More informationA.7.1 Trigonometric interpretation of dot product... 324. A.7.2 Geometric interpretation of dot product... 324
A P P E N D I X A Vectors CONTENTS A.1 Scling vector................................................ 321 A.2 Unit or Direction vectors...................................... 321 A.3 Vector ddition.................................................
More informationVectors. The magnitude of a vector is its length, which can be determined by Pythagoras Theorem. The magnitude of a is written as a.
Vectors mesurement which onl descries the mgnitude (i.e. size) of the oject is clled sclr quntit, e.g. Glsgow is 11 miles from irdrie. vector is quntit with mgnitude nd direction, e.g. Glsgow is 11 miles
More informationPure C4. Revision Notes
Pure C4 Revision Notes Mrch 0 Contents Core 4 Alger Prtil frctions Coordinte Geometry 5 Prmetric equtions 5 Conversion from prmetric to Crtesin form 6 Are under curve given prmetriclly 7 Sequences nd
More informationSection 7.1 Solving Right Triangles
Section 7.1 Solving Right Triangles Note that a calculator will be needed for most of the problems we will do in class. Test problems will involve angles for which no calculator is needed (e.g., 30, 45,
More informationMA 15800 Lesson 16 Notes Summer 2016 Properties of Logarithms. Remember: A logarithm is an exponent! It behaves like an exponent!
MA 5800 Lesson 6 otes Summer 06 Rememer: A logrithm is n eponent! It ehves like n eponent! In the lst lesson, we discussed four properties of logrithms. ) log 0 ) log ) log log 4) This lesson covers more
More informationGraphs on Logarithmic and Semilogarithmic Paper
0CH_PHClter_TMSETE_ 3//00 :3 PM Pge Grphs on Logrithmic nd Semilogrithmic Pper OBJECTIVES When ou hve completed this chpter, ou should be ble to: Mke grphs on logrithmic nd semilogrithmic pper. Grph empiricl
More informationModule 5. Three-phase AC Circuits. Version 2 EE IIT, Kharagpur
Module 5 Three-hse A iruits Version EE IIT, Khrgur esson 8 Three-hse Blned Suly Version EE IIT, Khrgur In the module, ontining six lessons (-7), the study of iruits, onsisting of the liner elements resistne,
More informationOr more simply put, when adding or subtracting quantities, their uncertainties add.
Propgtion of Uncertint through Mthemticl Opertions Since the untit of interest in n eperiment is rrel otined mesuring tht untit directl, we must understnd how error propgtes when mthemticl opertions re
More informationExample A rectangular box without lid is to be made from a square cardboard of sides 18 cm by cutting equal squares from each corner and then folding
1 Exmple A rectngulr box without lid is to be mde from squre crdbord of sides 18 cm by cutting equl squres from ech corner nd then folding up the sides. 1 Exmple A rectngulr box without lid is to be mde
More informationRight Triangles 4 A = 144 A = 16 12 5 A = 64
Right Triangles If I looked at enough right triangles and experimented a little, I might eventually begin to notice a relationship developing if I were to construct squares formed by the legs of a right
More informationDEFINITIONS. Perpendicular Two lines are called perpendicular if they form a right angle.
DEFINITIONS Degree A degree is the 1 th part of a straight angle. 180 Right Angle A 90 angle is called a right angle. Perpendicular Two lines are called perpendicular if they form a right angle. Congruent
More information9.3. The Scalar Product. Introduction. Prerequisites. Learning Outcomes
The Sclr Product 9.3 Introduction There re two kinds of multipliction involving vectors. The first is known s the sclr product or dot product. This is so-clled becuse when the sclr product of two vectors
More informationMath 135 Circles and Completing the Square Examples
Mth 135 Circles nd Completing the Squre Exmples A perfect squre is number such tht = b 2 for some rel number b. Some exmples of perfect squres re 4 = 2 2, 16 = 4 2, 169 = 13 2. We wish to hve method for
More informationFirm Objectives. The Theory of the Firm II. Cost Minimization Mathematical Approach. First order conditions. Cost Minimization Graphical Approach
Pro. Jy Bhttchry Spring 200 The Theory o the Firm II st lecture we covered: production unctions Tody: Cost minimiztion Firm s supply under cost minimiztion Short vs. long run cost curves Firm Ojectives
More informationLISTENING COMPREHENSION
PORG, přijímí zkoušky 2015 Angličtin B Reg. číslo: Inluded prts: Points (per prt) Points (totl) 1) Listening omprehension 2) Reding 3) Use of English 4) Writing 1 5) Writing 2 There re no extr nswersheets
More informationMATH 150 HOMEWORK 4 SOLUTIONS
MATH 150 HOMEWORK 4 SOLUTIONS Section 1.8 Show tht the product of two of the numbers 65 1000 8 2001 + 3 177, 79 1212 9 2399 + 2 2001, nd 24 4493 5 8192 + 7 1777 is nonnegtive. Is your proof constructive
More informationCumulative Test. 161 Holt Geometry. Name Date Class
Choose the best answer. 1. P, W, and K are collinear, and W is between P and K. PW 10x, WK 2x 7, and PW WK 6x 11. What is PK? A 2 C 90 B 6 D 11 2. RM bisects VRQ. If mmrq 2, what is mvrm? F 41 H 9 G 2
More informationRadius of the Earth - Radii Used in Geodesy James R. Clynch February 2006
dius of the Erth - dii Used in Geodesy Jmes. Clynch Februry 006 I. Erth dii Uses There is only one rdius of sphere. The erth is pproximtely sphere nd therefore, for some cses, this pproximtion is dequte.
More informationFactoring Polynomials
Fctoring Polynomils Some definitions (not necessrily ll for secondry school mthemtics): A polynomil is the sum of one or more terms, in which ech term consists of product of constnt nd one or more vribles
More informationChapter. Fractions. Contents: A Representing fractions
Chpter Frtions Contents: A Representing rtions B Frtions o regulr shpes C Equl rtions D Simpliying rtions E Frtions o quntities F Compring rtion sizes G Improper rtions nd mixed numers 08 FRACTIONS (Chpter
More informationIn order to master the techniques explained here it is vital that you undertake the practice exercises provided.
Tringle formule m-ty-tringleformule-009-1 ommonmthemtilprolemistofindthenglesorlengthsofthesidesoftringlewhen some,utnotllofthesequntitiesreknown.itislsousefultoeletolultethere of tringle from some of
More informationSPECIAL PRODUCTS AND FACTORIZATION
MODULE - Specil Products nd Fctoriztion 4 SPECIAL PRODUCTS AND FACTORIZATION In n erlier lesson you hve lernt multipliction of lgebric epressions, prticulrly polynomils. In the study of lgebr, we come
More informationParallel and Perpendicular. We show a small box in one of the angles to show that the lines are perpendicular.
CONDENSED L E S S O N. Parallel and Perpendicular In this lesson you will learn the meaning of parallel and perpendicular discover how the slopes of parallel and perpendicular lines are related use slopes
More information15.6. The mean value and the root-mean-square value of a function. Introduction. Prerequisites. Learning Outcomes. Learning Style
The men vlue nd the root-men-squre vlue of function 5.6 Introduction Currents nd voltges often vry with time nd engineers my wish to know the verge vlue of such current or voltge over some prticulr time
More information0.1 Basic Set Theory and Interval Notation
0.1 Bsic Set Theory nd Intervl Nottion 3 0.1 Bsic Set Theory nd Intervl Nottion 0.1.1 Some Bsic Set Theory Notions Like ll good Mth ooks, we egin with definition. Definition 0.1. A set is well-defined
More informationThe Triangle and its Properties
THE TRINGLE ND ITS PROPERTIES 113 The Triangle and its Properties Chapter 6 6.1 INTRODUCTION triangle, you have seen, is a simple closed curve made of three line segments. It has three vertices, three
More informationLearning Outcomes. Computer Systems - Architecture Lecture 4 - Boolean Logic. What is Logic? Boolean Logic 10/28/2010
/28/2 Lerning Outcomes At the end of this lecture you should: Computer Systems - Architecture Lecture 4 - Boolen Logic Eddie Edwrds eedwrds@doc.ic.c.uk http://www.doc.ic.c.uk/~eedwrds/compsys (Hevily sed
More informationWHAT HAPPENS WHEN YOU MIX COMPLEX NUMBERS WITH PRIME NUMBERS?
WHAT HAPPES WHE YOU MIX COMPLEX UMBERS WITH PRIME UMBERS? There s n ol syng, you n t pples n ornges. Mthemtns hte n t; they love to throw pples n ornges nto foo proessor n see wht hppens. Sometmes they
More informationVolumes by Cylindrical Shells: the Shell Method
olumes Clinril Shells: the Shell Metho Another metho of fin the volumes of solis of revolution is the shell metho. It n usull fin volumes tht re otherwise iffiult to evlute using the Dis / Wsher metho.
More information4.11 Inner Product Spaces
314 CHAPTER 4 Vector Spces 9. A mtrix of the form 0 0 b c 0 d 0 0 e 0 f g 0 h 0 cnnot be invertible. 10. A mtrix of the form bc d e f ghi such tht e bd = 0 cnnot be invertible. 4.11 Inner Product Spces
More informationSeeking Equilibrium: Demand and Supply
SECTION 1 Seeking Equilirium: Demnd nd Supply OBJECTIVES KEY TERMS TAKING NOTES In Setion 1, you will explore mrket equilirium nd see how it is rehed explin how demnd nd supply intert to determine equilirium
More informationIntro to Circle Geometry By Raymond Cheong
Into to Cicle Geomety By Rymond Cheong Mny poblems involving cicles cn be solved by constucting ight tingles then using the Pythgoen Theoem. The min chllenge is identifying whee to constuct the ight tingle.
More informationPythagorean Theorem: 9. x 2 2
Geometry Chapter 8 - Right Triangles.7 Notes on Right s Given: any 3 sides of a Prove: the is acute, obtuse, or right (hint: use the converse of Pythagorean Theorem) If the (longest side) 2 > (side) 2
More informationP.3 Polynomials and Factoring. P.3 an 1. Polynomial STUDY TIP. Example 1 Writing Polynomials in Standard Form. What you should learn
33337_0P03.qp 2/27/06 24 9:3 AM Chpter P Pge 24 Prerequisites P.3 Polynomils nd Fctoring Wht you should lern Polynomils An lgeric epression is collection of vriles nd rel numers. The most common type of
More information2 DIODE CLIPPING and CLAMPING CIRCUITS
2 DIODE CLIPPING nd CLAMPING CIRCUITS 2.1 Ojectives Understnding the operting principle of diode clipping circuit Understnding the operting principle of clmping circuit Understnding the wveform chnge of
More informationMath Review for Algebra and Precalculus
Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Mt Review for Alger nd Prelulus Stnley Oken Deprtment of Mtemtis
More informationReview. Scan Conversion. Rasterizing Polygons. Rasterizing Polygons. Triangularization. Convex Shapes. Utah School of Computing Spring 2013
Uth Shool of Computing Spring 2013 Review Leture Set 4 Sn Conversion CS5600 Computer Grphis Spring 2013 Line rsteriztion Bsi Inrementl Algorithm Digitl Differentil Anlzer Rther thn solve line eqution t
More informationVector differentiation. Chapters 6, 7
Chpter 2 Vectors Courtesy NASA/JPL-Cltech Summry (see exmples in Hw 1, 2, 3) Circ 1900 A.D., J. Willird Gis invented useful comintion of mgnitude nd direction clled vectors nd their higher-dimensionl counterprts
More informationReview Problems for the Final of Math 121, Fall 2014
Review Problems for the Finl of Mth, Fll The following is collection of vrious types of smple problems covering sections.,.5, nd.7 6.6 of the text which constitute only prt of the common Mth Finl. Since
More informationBasic Analysis of Autarky and Free Trade Models
Bsic Anlysis of Autrky nd Free Trde Models AUTARKY Autrky condition in prticulr commodity mrket refers to sitution in which country does not engge in ny trde in tht commodity with other countries. Consequently
More information50 MATHCOUNTS LECTURES (10) RATIOS, RATES, AND PROPORTIONS
0 MATHCOUNTS LECTURES (0) RATIOS, RATES, AND PROPORTIONS BASIC KNOWLEDGE () RATIOS: Rtios re use to ompre two or more numers For n two numers n ( 0), the rtio is written s : = / Emple : If 4 stuents in
More informationaddition, there are double entries for the symbols used to signify different parameters. These parameters are explained in this appendix.
APPENDIX A: The ellipse August 15, 1997 Becuse of its importnce in both pproximting the erth s shpe nd describing stellite orbits, n informl discussion of the ellipse is presented in this ppendix. The
More informationRIGHT TRIANGLE TRIGONOMETRY
RIGHT TRIANGLE TRIGONOMETRY The word Trigonometry can be broken into the parts Tri, gon, and metry, which means Three angle measurement, or equivalently Triangle measurement. Throughout this unit, we will
More informationClause Trees: a Tool for Understanding and Implementing Resolution in Automated Reasoning
Cluse Trees: Tool for Understnding nd Implementing Resolution in Automted Resoning J. D. Horton nd Brue Spener University of New Brunswik, Frederiton, New Brunswik, Cnd E3B 5A3 emil : jdh@un. nd spener@un.
More informationThe Cat in the Hat. by Dr. Seuss. A a. B b. A a. Rich Vocabulary. Learning Ab Rhyming
MINI-LESSON IN TION The t in the Ht y Dr. Seuss Rih Voulry tme dj. esy to hndle (not wild) LERNING Lerning Rhyming OUT Words I know it is wet nd the sun is not sunny. ut we n hve Lots of good fun tht is
More information