John L. Lehet
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1 New! ndroid pp! ST Mthemtics Review Geometry John L. Lehet ST Mth Diy Question ndroid pp - new question ech dy - rchive of over 200 questions - different eves nd subjects - for $1.99 on Googe Py or mzon Visit nd cick ST Mth Home for more informtion Fiing to Prepre is Prepring to Fi!
2 Geometric Nottion F F F F F The ine contining point nd F The ine segment with endpoints nd F The ength of ine segment F The ry strting t nd extending infinitey through F The nge formed by nd F m F The mesure of nge F F FG FG The tringe with vertices, nd F The qudriter with vertices,, F nd G is perpendicur to FG
3 Points nd Lines unique ine, contining points nd M M is the midpoint of, so M = M C 3 D 5 E CD = 3 nd DE = 5, so CE = 3+5 = 8 Probem 1:, nd C ie on the sme ine, if C is the midpoint of nd = 12, wht is C? Probem 2: On the ine bove, if CD = 4, EF = 2 nd CF = 10, wht is the vue of DE? C D E F
4 nges in the Pne Opposite nges formed by intersecting ines re equ nd re ced vertic nges Y Suppementry nges re Stright nges nd re equ to 180 degrees X Z W So, X = Z nd W = Y So, X +W = 180º, X+Y = 180º, W+Z = 180º, Y+Z = 180º Probem 1: In the bove digrm, if X is equ to 40 degrees, wht is the vue of W? Wht is the vue of Z? Wht is the vue of Y? c d b g e h f m If two pre ines ( nd m) re intersected by third ine, the ternte interior nges re equ for exmpe, e nd d re ternte interior nges Probem 2: In the bove digrm, nd m re pre, nme nges tht re equ to nge d? Nme nges tht re suppementry to nge b?
5 Tringes xº zº c Equiter Tringe equ sides (=b=c) equ nges (x=y=z=60) nges mesure 60º b b xº Isoscees Tringe two equ sides (=b) two equ nges (x=y) b Right Tringe one nge is 90º ( = 90) two sides re perpendicur 2 + b 2 = c 2 (Pythgoren Theorem) c Probem 1: If C is n Isoscees Tringe, such tht C = C nd m C is 40º, wht is the m C? Probem 2: If C is Right Tringe, such tht m C is 35º, wht is the m C if it is not 90º?
6 Tringes x 3 30º 2x x 45º x 2 3x 5x 60º 45º x x 4x 30º-60º-90º Tringe 45º-45º-90º Tringe Tringe Probem 1: If C is Right Tringe, such tht m C is 45º nd C = 4, wht is the ength of the ongest side?
7 Congruent Tringes tringes tht hve the sme size nd shpe b F E zº C = DEF b c = DE = xº xº C = EF = b zº C = DF = c c C D Probem 1: If C nd DEF re congruent tringes, nd =5 nd C=15, wht is EF? Simir Tringes tringes tht hve the sme shpe (corresponding nges re equ) E 8 xº 14 zº 10 C D 4 5 xº 7 zº F C nd DEF re simir tringes sides re proportion Probem 2: If C nd DEF re simir tringes, nd =5, C=7 nd DE=15, wht is EF?
8 8 xº Tringe Inequity The sum of the engths of ny two sides of tringe is greter thn the ength of the third side zº 11 C C < Probem 1: In C, = 3 nd C = 7, cn C be 4? Cn C be 12? Wht re the rnges of vues of C? Tringe Perimeter nd re Perimeter = b + + c (sum of the three sides) re = ½bh h c b = bse Probem 2: In the bove tringe, if =6, b= 4, c=7 nd h=5, wht is the perimeter? Wht is the re?
9 b C Qudriters w C C h xº s 2 s xº b D Preogrm opposite nges re equ opposite sides re equ P = 2 + 2b = bh w D Rectnge preogrm with right nges P = 2w + 2 = w C = D = ( 2 +w 2 ) D Squre rectnge with four equ sides P = 4s = s 2 C = D = s 2 Other Poygons Regur Poygon is poygon with sides nd nges equ Determine unknown nge nd sides using tringes sides interior nge sum 3 180º 4 360º 5 540º : : n 180(n-2)º
10 O D Circes O = Origin of Circe the center O = O = Rdius of Circe = rc C = Dimeter of Circe (twice the rdius O or O) C the ine segment D is tngent to the circe t point. D touches the circe t ony point. The Dimeter of circe is twice the Rdius of the circe d = 2r The Circumference of circe is the distnce round the circe it is nogous to perimeter of poygon C = d = 2 r The re of circe is the mount of spce within the circe = r 2 Probem 1: Given circe with center O nd re 16. Points nd re on the circe nd nge O is 30º. Find the ength of ine segment. O 30º
11 Soid Figures h s h w Rectngur Soid Think of Crdbord ox V = wh S = 2w + 2h + 2wh s Cube Speci Rectngur Soid in which =w=h=s V = s 3 S = 6s 2 s Cyinder Think of cn of soup V = r 2 h S = 2 r rh Cyinder Unfoded Rectngur Soid Unfoded six rectnges Cube Unfoded six squres circe on top circe on bottom, rectnge in the midde
12 Soid Figures Sphere Think of b rdii re equ r h Cone V = (1/3) r 2 h Its Voume is 1/3 of cyinder with the sme height nd bse r h Pyrmid squre t the bse with four tringes V = s 2 h/3 Pyrmid Unfoded Probem 1: If the voume of cube is 125 in 3, wht is the ength of side? Wht the Surfce re of the cube? Probem 2: If two cyinders hve equ voume nd the ter is four times higher thn the shorter, wht is the rtio of the rdii? squre in the midde, with four congruent tringes
13 Coordinte Geometry Positive Sope y = 2x - 1 Negtive Sope Zero Sope y = -2x + 2 y = 5 Two ines re pre when their sopes re the sme y = 2x + 3 is pre to y = 2x 7 since the sope of both ines is 2 Undefined Sope x = 4 Two ines re perpendicur when their sopes re negtive reciprocs OR the product of the sopes is -1 y = -2x + 3 is perpendicur to y = (1/2)x 7 since (-2)(1/2) = -1 OR (-2) is the negtive reciproc of (1/2) Probem 1: Give ine tht is pre to the ine y = 3x 4. Give ine tht is perpendicur to it. Midpoint Formu Given two points (x 1, y 1 ) nd (x 2, y 2 ) their midpoint is (x m, y m ) where x m = (x 1 +x 2 )/2 nd y m = (y 1 +y 2 )/2 Distnce Formu Given two points, (x 1, y 1 ) nd (x 2, y 2 ) their distnce is d where d = [(x 1 -x 2 ) 2 +(y 1 -y 2 ) 2 ] Probem 2: Wht is the distnce of the two points (1,4) nd (-1,-2)? Wht is their midpoint? Probem 3: If (3,2) is the midpoint of two points, one being (-1,-2), wht is the other point?
14 Trnsformtions The tringe ws rotted cockwise 55º The tringe ws refected bout the y-xis The tringe ws trnsted up nd right Trnstion Moves up/down nd eft/right Rottion Rottes on point Not necessriy the center counter- cockwise cockwise Refection Refects ong ine of symmetry Probem 1: If cock is rotted 90 degrees cockwise, wht number wi be t the top? (-2,3) Probem 2: If the tringe to the right is refected bout the y-xis, wht re the new co-ordintes? If refected bout the x-xis, wht re the new co-ordintes? Probem 3: If the tringe to the right is trnsted 2 units up nd 3 units eft, wht re the new co-ordintes? If then (fter trnstion) it is refected bout the x-xis, wht re the new co-ordintes? (-4,-1) (3,-1)
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