Geometry 71 Geometric Mean and the Pythagorean Theorem


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1 Geometry 71 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 $8,000 question nd the $32,000 question on Who Wnts to e Millionire?. Ex 2: Find the geometric men etween 2 nd 10.. Theorem 71 If the ltitude is drwn from the vertex of the right ngle of tringle to its hypotenuse, then the two tringles formed re similr to the given tringle nd to ech other. D V : V : V. Theorem 72 The mesures of the ltitude drwn from the vertex of the right ngle of right tringle to its hypotenuse is, the geometric men etween the mesures of the two segments of the hypotenuse. w D x V D : V D D D = D D w = x Ex 3: Find the length of the ltitude, if the following is true. 6 20
2 D. Theorem 73 If the ltitude is drwn to the hypotenuse of right tringle, then the mesure of leg of the tringle is the geometric men etween the mesures of the hypotenuse nd the segment of the hypotenuse djcent to tht leg. w D x V : V D : V D D = w = x + w ( hyp) Ex 4: Find the length of the given sides if the following is true. 4 y x 6 z HW: Geometry 71 p ll, ll, 4243, 4950, odd Hon: 34, 44,
3 Geometry 72 The Pythgoren Theorem nd its onverse. Theorem Pythgoren Theorem In right tringle, the sum of the squres of the mesures of the legs equls the squre of the mesure of the hypotenuse. c + = c Ex 1: Find x. 7 x 14. Theorem onverse of the Pythgoren Theorem If the sum of the squres of the mesures of two sides of tringle equls the squre of the mesure of the longest side, then the tringle is right tringle. 1. Pythgoren Triple is whole numers tht stisfy the eqution + = c. Ex 2: Determine if the mesures of these sides re the sides of right tringle. 40, 41, 48 HW: Geometry 72 p odd, 3035, 40, 4647, odd, odd Hon: 39
4 Geometry 73 Specil Right Tringles. 45 o 45 o 90 o Tringles x d Do the Pythgoren Theorem (solve for d) = c 2 2 x + x 2 = d x 1. Theorem In 45 o 45 o 90 o tringle, the length of the hypotenuse is 2 times s long s leg. s s 2 leg 2 = hypotenuse s Exmple 1: Find the length of the sides of the tringle. 6 Exmple 2: If the leg of 45 o 45 o 90 o tringle is 12 units, find the length of the hypotenuse.. 30 o 60 o 90 o Tringles 1. Wht is the reltionship etween the short leg of 30 o 60 o 90 o, tringle nd the hypotenuse? short leg ( ) = hypotenuse 2. Let s do Pythgoren Theorem to solve for. + x = (2 x) 60 o 60 o x D x short leg ( ) = long leg
5 1. Theorem In 30 o 60 o 90 o tringle, the length of the hypotenuse is twice the length of the short leg, nd the length of the long leg is 3 times the length of the short leg. n 3 30 o 2n 60 o n Exmple 3: Find nd. 60 o 12 Exmple 4: V WXY is 30 o 60 o 90 o tringle with right ngle X nd WX s the longer leg. Grph points X (2, 7) nd Y(7, 7), nd locte point W in qudrnt III. HW: Geometry 73 p , 27, 29, 36, 37, 40, 4344, odd Hon: 26, 38
6 74 Trigonometry Rtios in Right Tringles. Rtios 1. Trigonometry helps us solve mesures in right tringles.. Trigon mens tringle. Metron mens mesure. Tringle mesures revition Definition sin leg opposite = hypotenuse c c cos leg djcent to = hypotenuse c tn leg opposite to = djcent Exmple 1: Find the sin S, cos S, tn S, sin E, cos E, tn E. M 8 6 S 10 E Ex 2: Solve the tringle 6 35 o
7 Ex. 3: Solve the tringle 10 X Y 4 Y Ex 4: plne is one mile ove se level when it egins to clim t constnt ngle of 2 o for the next 70 ground miles. How fr ove se level is the plne fter its clim? 1 mile HW: Geometry 74 p , 6364, odd Hon: 5558, 6568
8 Geometry 75 ngles of Elevtion nd Depression. Definitions: 1. n ngle of elevtion is the ngle where if you strt horizontl nd move upwrd. ngle of elevtion 2. n ngle of depression is the ngle where you strt horizontl nd move downwrd. ngle of depression Ex 1: mn stnds on uilding nd sees his friend on the ground. If the uilding is 70 m tll nd the ngle of depression is 35 o, how fr is the mn from the uilding? Ex 2: mn notices the ngle of elevtion to the top of tree is 60 o, if he is 14 m from the tree, how tll is the tree? HW: Geometry 75 p , 9, 11, 13, 1418, 2829, odd, 3639, odd Hon: 19, 24
9 Geometry 76 The Lw of Sines. The Lw of Sines  In trigonometry, the Lw of Sines cn e used to find missing prts of tringles tht re not right tringles. 1. Let V e ny tringle with sides,, nd c representing the mesures of the sides opposite the ngles with mesures, nd respectively. sin sin sin Then = =. c c 2. Proof of Lw of Sines Given: D is n ltitude of Prove: sin = sin V. Sttements Resons 1.) D is n ltitude of V 1.) 2.) V D nd VD re rt V s. 2.) Def of rt V s. h h 3.) sin = nd sin = 3.) Def of sine 4.) (sin ) = h nd h = (sin ) 4.) 5.) (sin ) = (sin ) 5.) sin sin 6.) = 6.) Multiply ech side y h D Exmple 1: Find p. Round to the nerest tenth. 8 Q P 17 o 29 o R Exmple 2: Solve V DEF if m D = 112 o, m F = 8 o, nd f = 2 Round to the nerest tenth.
10 HW: Geometry 76 p odd, 3839, 43, Hon: 4445
11 Geometry 77 The Lw of osines. The Lw of osines  The Lw of osines llows us to solve tringle when the Lw of Sines cnnot e used. 1. Let V e ny tringle with sides,, nd c representing the mesures of the sides opposite the ngles with mesures, nd respectively. Then the following equtions re true: = + c 2c cos = + c 2ccos c = + 2cos c 2. You cn use the Lw of osines when you know two sides nd the included ngle. o 3Exmple 1: Find c if = 8, = 6, nd + 48 c = + 2cos 48 o 8 6 c 3. You cn use the Lw of osines when you know ll three sides nd re looking for n ngle. Exmple 2: Use the Lw of osines to solve for = + c 2c cos HW: Geometry 77 p odd, 42, 4647, odd Hon: 39, 43, 57, 59
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