Geoetry Perieter d re Trigle Retgle Squre h Perieter: P = + + re: = h l Perieter: P = l + w w re: = lw s Perieter: P = 4s re: = s Trpezoid h d Prllelogr Perieter: P = + + + d h re: = h( + ) Perieter: P = + re: = h Cirle d r Ciruferee: C = πr = πd re: = πr gles Clssified y Mesure. ute 0 < < 90. Right = 90 3. Otuse 90 < < 80 4. Stright = 80 Ters Desriig gles Copleetry gles: Two gles re opleetry if the su of their esures is 90. Suppleetry gles: Two gles re suppleetry if the su of their esures is 80. Cogruet gles: If two gles hve the se esure, they re sid to e ogruet gles (syolized s ). Vertil gles: Vertil gles re ogruet. Tht is, vertil gles hve the se esure. djet gles: Two gles re djet if they hve oo side. Trsversl: lie i ple tht itersets two or ore lies i tht ple i differet poits. If two prllel lies re ut y trsversl, the the followig two stteets re true.. Correspodig gles re ogruet.. lterte iterior gles re ogruet. The Pythgore Theore I right trigle, the squre of the legth of the hypoteuse is equl to the su of the squres of the legths of the two legs: = +. 90
Reltioships Betwee Mesureets i the U.S. Custory Syste ihes (i.) = foot (ft) 36 ihes = yrd Legth Weight 3 feet = yrd (yd) 580 feet = ile (i) 6 oues (oz) = poud (l) 000 pouds = to (T) Cpity 8 fluid oues (fl oz) = up () ups = pit (pt) = 6 fluid oues Tie 60 seods (se) = iute (i) 60 iutes = hour (hr) pits = qurt (qt) 4 qurts = gllo (gl) 4 hours = dy 7 dys = week Reltioships Betwee Mesureets i the Metri Syste Legth illieter () 00 eter = 000 etieter () 0 eter = 00 deieter (d) eter = 0 d eter () =.0 eter deketer (d) = 0 eters hetoeter (h) = 00 eters kiloeter (k) = 000 eters Cpity (Liquid Volue) illiliter (L) 00 liter L = 000 L liter (L) =.0 liter hetoliter (hl) = 00 liters kiloliter (kl) = 000 liters L 00 kl Mss illigr (g) 00 gr g = 000 g etigr (g) 0 gr deigr (dg) gr gr (g) =.0 gr dekgr (dg) = 0 grs hetogr (hg) = 00 grs kilogr (kg) = 000 grs g 00 kg etri to (t) = 000 kilogrs kg 00 t t = 000 kg =,000,000 g =,000,000,000 g Lrge Uits of Mesureet i the Metri Syste Meg- Gig- Ter- Celsius egyte = 000 000 ytes gigyte = 000 000 000 ytes teryte = 000 000 000 000 ytes U.S. Custory d Metri Equivlets U.S. to Metri i.. 54 ext ft 305 yd = 094. i =. 6 k = ( ) Legth Metri to U.S. 394 i. = 3. 8 ft =. 09 yd k 6 i Cpity (Liquid Volue) qt 946 L gl = 3. 785 L L =. 06 qt L 64 gl L = ui etieter ( or 3 ) Weight (Mss) oz = 835. g l 454 kg g 035 oz kg =. 05 l Teperture Equivlets Fhreheit 00 Wter oils 90 94 80 76 70 58 60 40 50 40 04 30 Cofort 86 0 rge 68 0 50 0 Wter freezes 3
Foruls d Defiitios P For the Proportio 00 = B P% = peret (writte s the rtio P 00 ) B = se (uer tht we re fidig the peret of) = out ( prt of the se) Ters Relted to Sles Profit: The differee etwee sellig prie d ost. profit = sellig prie - ost Peret of Profit: profit. Peret of profit sed o ost: ost profit. Peret of profit sed o sellig prie: sellig prie profit 3. Peret of profit sed o oey ivested: oey ivested Disout: redutio i the origil sellig prie of ite disout = origil prie - sellig prie Sles Tx: tx hrged o the tul sellig prie of goods sold y retilers Coissio: fee pid to get or slesperso for servie Forul for Clultig Siple Iterest I = P r t, where I = iterest (ered or pid) P = priipl (the out ivested or orrowed) r = rte of iterest (stted s ul rte) i deil or frtio for t = tie (oe yer or frtio of yer) List of Coo Foruls Forul Meig 5 C = ( F 3 ) Teperture i degrees Celsius 9 C equls 5 ties the differee etwee the Fhreheit 9 teperture F d 3. d = rt The diste trveled d equls the produt of the rte of speed r d the tie t. L = πrh The lterl surfe re L (top d otto ot iluded) of ylider is equl to π ties the rdius r of the se ties the height h. F = I physis, the fore F tig o ojet is equl to its ss ties elertio. + + g = 80 The su of the gles of trigle (,, d g) is 80. Ters Relted to Futios Reltio: set of ordered pirs of rel uers Futio: reltio i whih eh doi eleet hs extly oe orrespodig rge eleet Doi of Reltio: The set of ll first oordites i reltio Rge of Reltio: The set of ll seod oordites i reltio
Ftorig Polyoils Speil Ftorig Tehiques. x = ( x+ ) ( x ): differee of two squres. x + x+ = x+ 3. x x+ = x ( ) : squre of ioil su ( ) : squre of ioil differee Iequlities Lier Iequlities Lier iequlities hve the followig fors where,, d re rel uers d 0: x + < d x + x + > d x + Copoud Iequlities The iequlities < x + < d d x + d re lled opoud lier iequlities. (This iludes < x+ d d x+ < d s well.) Itervl Nottio Type of Itervl lgeri Nottio Itervl Nottio Ope Itervl < x < (, ) Grph Closed Itervl x, Hlf-ope Itervl Ope Itervl x< < x x> x<, ) (, (, ) (, ) Hlf-ope Itervl x x, ) (, Qudrti Equtios Qudrti Forul The solutios of the geerl qudrti equtio x + x + = 0, where 0, re x = ± 4. Geerl Ifortio o Qudrti Futios For the qudrti futio y= x + x+. If > 0, the prol opes upwrd.. If < 0, the prol opes dowwrd. 3. x = is the lie of syetry. 4. The vertex (turig poit) ours where x =. Vertex Sustitute this vlue for x i the futio d fid the y-vlue of the vertex. The vertex is the lowest poit (iiu vlue) o the urve if the prol opes upwrd or it is the highest poit (xiu vlue) o the urve if the prol opes dowwrd. y Lie of Syetry x
Rules Rules for Multiplitio d Divisio with Itegers. Whe the sigs re like, the produt or quotiet is positive.. Whe the sigs re ot like, the produt or quotiet is egtive. Rules for Order of Opertios. First, siplify withi groupig syols, suh s pretheses ( ), rkets [ ], res { }, rdil sigs, solute vlue rs,or frtio rs. Strt with the ierost groupig.. Seod, evlute y uers or expressios rised to expoets. 3. Third, ovig fro left to right, perfor y ultiplitio or divisio i the order i whih it ppers. 4. Fourth, ovig fro left to right, perfor y dditio or sutrtio i the order i whih it ppers. Divisiility Rules If the uits digit of whole uer is 0,, 4, 6, or 8 ( eve digit), the the uer is divisile y. If the su of the digits of whole uer is divisile y 3, the the uer is divisile y 3. If the uits digit of whole uer is 0 or 5, the the uer is divisile y 5. If the su of the digits of whole uer is divisile y 9, the the uer is divisile y 9. If the uits digit of whole uer is 0, the the uer is divisile y 0. Rules for Expoets For ozero rel uers d d itegers d : The expoet : = The expoet 0: 0 = The produt rule: = The quotiet rule: = + Negtive expoets: ( ) = Power rule: = ( ) = Power of produt: Power of quotiet: = Lier Equtios Sury of Foruls d Properties of Lies Stdrd for: x + By = C where d B do ot oth equl 0 Slope of lie: y y = where x x x x Slope-iterept for: y = x + with slope d y-iterept (0,) Poit-slope for: y y = x ( x ) with slope d poit ( x, y ) o the lie Horizotl lie, slope 0: y = Vertil lie, udefied slope: x =. Prllel lies hve the se slope.. Perpediulr lies hve slopes tht re egtive reiprols of eh other. Crtesi Coordite Syste Qudrt II (x egtive, y positive) (-, +) Qudrt III (x egtive, y egtive) (-, -) y-xis Qudrt I (x positive, y positive) (+, +) Qudrt IV (x positive, y egtive) (+, -) x-xis
Nottio d Syols Types of Nuers Nturl Nuers (Coutig Nuers): N = {,, 3, 4, 5, 6,...} Whole Nuers: W = { 0,,, 3, 4, 5, 6,...} Itegers: Z = {..., 4, 3,,, 0,,, 3, 4,...} Rtiol Nuers: rtiol uer is uer tht e writte i the for of where d re itegers d 0. Irrtiol Nuers: irrtiol uer is y uer tht e writte s ifiite orepetig deil. Rel Nuers: The rel uers osist of ll rtiol d irrtiol uers. Equlity d Iequlity Syols = is equl to is ot equl to solute Vlue < is less th > is greter th The solute vlue of rel uer is its diste fro 0. Syolilly, = if is positive uer or 0. = if is egtive uer. is less th or equl to is greter th or equl to Priiples d Properties Properties of dditio d Multiplitio For dditio Ne of Property For Multiplitio + = + Couttive Property = ( + )+ = + + ( ) ssoitive Property ( ) = ( ) + 0 = 0 + = Idetity = = + ( )= 0 Iverse = 0 ( ) Zero-Ftor Lw: 0 = 0 = 0 Distriutive Property: ( + )= + dditio Priiple of Equlity = B d + C = B + C hve the se solutios (where, B, d C re lgeri expressios). Multiplitio (or Divisio) Priiple of Equlity = B d C = BC d B = hve the se C C solutios (where d B re lgeri expressios d C is y ozero ostt, C 0). Zero-Ftor Property If the produt of two (or ore) ftors is 0, the t lest oe of the ftors ust e 0. Tht is, for rel uers d, if = 0, the = 0 d/or Properties of Squre Roots If d re positive rel uers, the. =. = Properties of Rdils If is positive iteger, is y iteger, d is rel uer, the. = = ( ) = ( ). or i rdil ottio, = ( ) =