Answers. Answers. 1 Consumer arithmetic

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1 Answers Consumer arithmetic Eercise. a $, b $, c $, d $, e $, f $, g $., h $. a $, b $ a $, b $., c $., d $9.9 a $, b $, c $., d $. a $9., b $., c $., d $. a $., b $., c $., d $. a $9., b $. a $, b $ 9a $, b $, c $ $. $ $ ears 9 months % p.a..% p.a. a % p.a., b % p.a., c % p.a., d % p.a. months 9 das Eercise. a Year Opening balance Interest Closing balance st $ $. $ nd $ $. $ rd $ $. $ 9. b $9. a Balance = $9.; Interest = $., b Balance = $.; Interest = $. a $., b $. a $., b $., c $9.9, d $.9, e $., f $.9, g $., h $. a $9., b $.9, c., d $9., e $.9, f $99., g $9., h $. a $., b $., c $., d $.9 a $., b $9.9, c $.9, d $9., e $9., f $., g $., h $., i $. a $ 9., b $9. 9 $ 9. a $.9, b $.9 Christine, $. B A, $. $. Value = $.; Interest = $. a $9, b $, c $ 9 a $., b $., c $., d $ $ $ $ a $, b $, c $, d $ Eercise. $ a $9, b $9, c $, d $ $ $ $ a $, b $ 9a kl, b 9 kl 9 a $ 9, b $ $9 ears ears ears $ $ % 9 a 9%, b $ Eercise. $. a $., b $. a $, b $ a no, b $. $9. a $, b $ a $, b.% 9 $ B, $ a $, b $9, c $9. a $ 9, b $.9 a $, b $, c $., d $., e $9. a $, b $, c $9., d $ 9., e $. a $, b $, c % 9.% 9.% Eercise. a April, b, c.9%, d $, e $9, f $., g $., h $., i $, j $. a.% p.a., b.% p.a., c.% p.a., d.% p.a., e.% p.a., f.9% p.a. a $., b $., c $., d $. a $., b $., c $., d $9. $. a $.9, b $9.9 a.%, b, c $.9 a $., b $., c $9.9 9a $, b $. a Yes, b $, b $. $. Answers

2 Mathscape Etension Eercise. a $., b $., c $., d $.9, e $., f $. a $., b $9., c $., d $., e $., f $. ai $, ii $, bi $9., ii $9., ci $, ii $, di $9., ii $. a $, b $, c $, d.% a $, b $, c.9% p.a. a $, b.%, c.% p.a. a Sunshine Bank, $ $ 9 $ a $, b $ 9, c.9% p.a. a No, a month is longer than fortnights. b -- ears, c $, d The loan is paid off more quickl and the amount of interest paid is reduced. ai $9, ii $9 9., iii $9., iv $ 99., b Yes, the repament is greater than the monthl interest. c No, the repament is less than the monthl interest. a $., b Increase the size of the repaments. Chapter Review a $, b $., c $9., d $. $ a $, b $.9, c $. a $., b $. $ % p.a. a $, b $, c $. Balance = $9.9, Interest = $9.9 9 a $., $., b $9., $9., c $., $., d $9., $., e $., $9., f $., $. a $., b $., c $., d $. a $., b $., c $.9, d $ 9., e $9., f $. a $ 9., b.% p.a. B, b $. A, b $. $ $. a $, b $, c $ a $, b $, c $, d $9 9 $ a $, b $ 9 ears a $, b no a $, b $, c.% p.a. a $9, b $ 9, c $ a $, b $, c $, d $. a $9, b $9, c % p.a. a $., b $. $. 9 a $., b $., c $., d $. a $9., b $., c $ a $, b $, c $, d.% p.a. $9 a $, b $ 9, c $.% p.a. Trigonometr Eercise. a opp = QR, adj = PQ, hp = PR, b opp = EG, adj = FG, hp = EF, c opp = XZ, adj = XY, hp = YZ ai --, ii --, iii --, bi -----, ii -----, iii -----, ci -----, ii -----, iii -----, di -----, ii -----, iii -----, 9 9 ei -----, ii -----, iii -----, fi -----, ii -----, iii aα, b θ, c θ, d θ, e α, f α a -----, b -----, c -----, d -----, e -----, f -----, g -----, h -----, i -----, j -----, k -----, l a =, b = 9, c p =, d c =, e e =, f n = a sin θ = -----, cos θ = -----, tan θ = a cm, b sin θ = -----, cos θ = -----, 9 tan θ = a mm, b sin α = -----, cos α = -----, tan α = a sin θ = -----, cos θ = -----, b cos θ = -----, tan θ = -----, c tan θ = -----, sin θ = sin θ = , cos θ = Eercise. a, b, c, d, e, f, g, h a, b, c, d, e, f, g 9 h a., b., c., d., e., f., g 9., h., i., j., k., l. a.9, b., c. a.9, b. a, b, c, d, e, f, g, h, i 9, j, k, l a, b 9, c, d a.,., b.9,. 9a 9, b 9, c Eercise. ap =., b m =., c =., d =., e t =., f a =., g g =., h b =., i k =., j z =., k c =., l w =., m e =., n n =., o u =.9 a =., b t =., c e =., d m = 9., e c =., f v =. a. cm, b. m, c. cm, d cm m

3 Answers. m. m m 9 cm 9. m. cm a. cm, b cm a cm, b cm a. cm, b. cm 9 m. m. m m. m 9. m m a m, b 9 m a mm, b. mm, c mm, d mm.9 cm a. cm, b. cm,, c. cm a cm, b cm m a AD =. cm, DC =. cm, b cm a.9 cm, b 9.9 cm 9 m Eercise. a, b, c, d, e, f 9, g, h 9, i, j 9, k, l a, b, c, d, e, f, g, h 9, i 9 a 9, b, c a, b, c 9 9 a 9, b a, b 9, c, d 9 9 a, b 9, c, d Eercise. 9 a sin θ = --, cos θ = --, tan θ = --, b sin θ = -----, cos θ = -----, tan θ = -----, c sin θ = -----, cos θ = -----, tan θ = a, b, c, d a -----, b -----, c -----, d a, b, c, d, e 9, f a, b, c Eercise. a b b a ai --, ii --, iii --, iv --, bi sin θ = cos (9 θ), ii cos θ = sin (9 θ) a 9, c Yes, sin θ = cos (9 θ). c c c c a sin =., b cos =., c sin =.9, d cos =.9 a =, b =, c =, d =, e =, f =, g =, h =, i =, j =, k = 9, l = a, b, c, d a =, b =, c =, d =, e =, f =, g =, h =, i =, j =, k =, l = a., b. 9 a, b tan θ, c, d tan θ, e sin θ, f cos θ, g sin θ, h cos θ, i cos θ -- Eercise. a --, b , c , d , e , f, g , h --, i a, b --, c, d --, e --, f -- a --, b --, c, d --, e , f , g , h , i --, j --, k --, l a, b , c, d, e , f , g , h , i -- a , b , c , + + d a, b --, c, d --, e, f -- a, b, c --, d, e, f , g , h a, b, c, d, e, f, g, h, i a =, b k =, c t =, d n =, e u =, f e = 9 a h =, b p =, c w = a, b, c, d, e, f a cm, b cm a =, b =, c cm a cm, b cm m Eercise. aia = N E, B = S E, C = N W, ii A =, B =, C = 9, bia = S E, B = S W, C = N9 W, ii A =, B =, C =, cia = N E, B = S W, C = N W, ii A =, B =, C = a, b, c, d a 9, b a.9 km, b.9 km, c. km, d 9. km a 9. km, b. km, c. km, d. km ai 9, ii 9, bi, ii, ci 9, ii 9, di, ii ai HSP = + = 9, ii m, bi HBS = + = 9, ii. km a km, b 9a. NM, b. NM, c 9 NM, d 9 km

4 Mathscape Etension Chapter Review a -----, b -----, c -----, d -----, e -----, f = sin θ = -----, cos θ = a, b. aa = 9.9, b p =., c z =. a. cm, b. cm 9a, b, c a. cm, b. cm, c m. m m a cm, b 9 cm C 9 a sin θ, b cos θ a, b, c 9 a =, b =, c =, d =, e =, f = a, b tan θ, c cos θ, d sin θ a , b --, c, d , e --, f --, g, h a =, b = ai N9 E, ii 9, bi S E, ii, ci S W, ii, di N W, ii a 9, b 9 a 9. km, b.9 km a. NM, b.9 km ai, ii, bi, ii a IJK = + = 9, b m a PQR = 9 + = 9, b km Volume and surface area Eercise. a cm, b cm, c.9 cm, d. cm a cm, b cm, c cm. cm a cm, b cm, c.9 cm a 9 cm, b cm a cm, b cm ai = 9., ii cm, bih =, ii 9 cm a cm, b cm, c cm 9a =, b cm a.9 m, b $99. a. m, b $. cm, cm, cm 9 cm cm a cm, b cm, c cm d cm Eercise. a cm, b cm a cm, b cm, c cm b cm a cm, b cm, c cm a. cm, b. cm, c. cm a. cm, b 9. cm a 9. cm, b. cm b 9 cm, c. cm Eercise. a. cm, b. cm, c. cm, d 9. cm a π cm, b 9π cm a. cm, b 9. cm, c 99. cm, d.9 cm, e. cm, f. cm a 9. cm, b. cm, c cm a cm, b π cm, c π cm, d 9. cm a. cm, b 9. cm aa = πr + πrh, b mm a 9 cm, b cm 9a. m, b $. a 9π m, b π m, c.π m, d. m $. a 9 cm, b cm, c cm, d 9 cm cm π cm Eercise. a π cm, b π cm, c π cm a. cm, b. cm, c. cm, d. cm, e 9. cm, f. cm a cm, b cm, c cm, d 99 cm, e cm, f cm a π cm, b π cm, c π cm, d π cm a. cm, b. cm, c. cm a 9π cm, b π cm a 9 cm, b cm, c cm a. cm, b.9 cm, c. cm 9a = 9, S.A. = cm, b =, S.A. = cm, c = 9.9, S.A. = cm km a cm, b 9 cm a cm, b π cm cm : a cm, b 9 cm, c cm, d cm, e cm, f cm 9π cm Eercise. a units, b units, c 9 units a cm, b cm, c cm, d 9 cm, e 9. cm, f. cm a cm, b cm, c cm a cm, b cm, c 9 cm, d cm a. m, b. m, c. m a cm, b. cm, c 9. cm, d. cm a m, b 9 m, c m, d m, e. m, f. m a cm, b cm, c cm 9a 9 cm, b mm a mm, b m, c. cm a cm, b m, c mm a 9 cm, b mm, c. cm, d. m ai =., ii 9. cm, bip =, ii. cm, cik =, ii 9. cm a m, b m, c m, d. m, e. m, f m, g. m, h 9 m, i m a cm, b cm units

5 Answers Eercise. a cm, b cm, c 9 cm, d cm, e cm, f cm a. cm, b 9. cm, c. cm, d 9. cm a π mm, b π mm, c π mm, d π mm a 9 cm, b 9 cm, c 9. cm, d. cm, e. cm, f. cm a m, b m a m, b m a cm, b cm, c 9 cm, d 9 cm, e 9 cm, f 99 cm a π cm, b 9 cm, c.% 9ah = cm, r = cm, b 9 cm, c.9 L. L a.9 m, b 9 L cm a π cm, b π cm Eercise. a cm, b cm, c cm, d cm, e. cm, f. cm a cm, b. cm, c cm a. cm, b m, c mm, d. cm, e. m, f. mm. cm a cm, b cm a cm, b cm, c cm, d cm a. cm, b. cm a cm, b cm 9a cm, b cm, c cm a. cm, b. cm, c cm a cm, b. cm, c cm, d. cm, e. cm, f cm. cm a cm, b =, c cm a OA = 9 cm, OB = cm, b cm, c cm, d -- Eercise. a. cm, b. cm, c. cm, d. cm, e. cm, f 9. cm a cm, b cm, c cm, d cm a π cm, b π cm, c π cm ai =, ii.9 cm, bi =, ii. cm, ci =, ii. cm a mm, b mm a mm, b mm ai cm, ii cm, bi cm, ii cm a --, b cm 9. cm a cm, b. cm a cm, b 9 cm, c 9 cm, d cm, e cm, f cm ai cm, ii 9 cm, b cm a. L, b. L. cm. cm Eercise.9 a cm, b cm, c cm, d cm, e cm, f cm a. mm, b. mm c. mm, d.9 mm a π cm, b 9π cm, c π cm, d π cm, e π cm, f π cm a m, b m, c m a cm, b cm, c cm, d cm, e cm, f cm g a 9π cm, b π cm, c. cm a cm, b cm 9 : a cm, b cm, c cm. cm a cm, b ml a r cm, b cm, c -- πr cm, d :, e π cm Chapter Review a cm, b cm cm a =, b cm cm a PE = cm, PF = cm, b 9 cm cm cm cm 9. cm. cm 9. cm a π cm, b cm. cm cm cm a. cm, b. cm cm a cm, b cm 9 a cm, b cm, c cm. mm a w =, b cm, c L.9 cm a 9 cm, b cm a cm, b.9 cm a 99 cm, b 9 cm, c cm, d cm a π cm, b π cm, c π cm a 9 cm, b.9 cm a cm, b π cm 9 a cm, b cm a. cm, b cm, c cm a 9 cm, b cm a cm, b cm, c 9 cm, d 9 cm Deductive geometr Eercise. a =, b =, c t =, d u =, e p =, f k =, g m =, h a =, i z =, j r =, k w =, l n = a =, b p =, c r =, d k = 9, e =, f f = a =, b p =, c c =, d a =, e t =, f h =, g q = 9, h k =, i s = az =, b b =, c n =, d a =, e u =, v =, f =, =, g p = 9, h v =, w =, i k =, j d =, e =, k r = 9, s =, l j =, k =, m a = 9, b =, n f =, o = 9, z = a =, =, b p =, q =, c a =, b =, d g =, e k =, f n =, g c =, h b = 9, i v = 9 aa = 9, b =, b u = 9, v =, c c =, d =,

6 Mathscape Etension d m =, n =, e j =, k =, f g =, h =, i = at =, b n =, c r =, d c =, e h =, f k = a es, b no, c no 9 aa =, b =, c =, b =, =, z =, c p =, q =, r = 9 a b =, b e =, c g =, d =, e t =, f n =, g s =, h w =, i = a a =, b =, b m = 9, n = 9, c p =, q =, d =, =, e c =, d =, f g = 9, h = 9, g r =, s =, h u = 9, v =, i j =, k =, j e =, f =, k p =, q =, l m =, n = a a =, b =, b e =, f =, c p = 9, q =, d u = 9, v =, e g = 9, h = 9, f v =, w =, g j =, k =, h c =, d =, i s =, t = a m =, n =, b b = 9, c =, c a = 9, b =, c =, d =, d a =, b =, c =, e =, =, z =, f = 9, =, z = = = Eercise. a, b, c, d, e, f, g 9, h, i a conve, b non-conve, c conve, d non-conve a regular, b irregular, c irregular, d regular ai rectangle, ii rhombus, b no all sides and all angles are equal in regular polgons. a, b, c, d c = 9a, b, c a, b, c, d 9 a, b Eercise. a AED = (co-interior s, AB CD) CEF = (verticall opposite s) = c XZV = (corresponding s, TU VW) VZS = (adjacent s in a right angle) = e KLN = (co-interior s, KL MN) KLJ = ( s at a point) IJL = (co-interior s, IJ KL) = g BFG = (verticall opposite s) EGD = (co-interior s, AB CD) HGD = (HG bisects EGD) = i WRS = (corresponding s, PQ RS) TRP = (WV bisects TRS) VRU = (verticall opposite s) = b PRQ = (verticall opposite s) PQR = ( sum of a ) PQU = (UQ bisects PQR) = d EFG = (co-interior s, EF HG) DEF = (alternate s, DE FG) EDF = ( sum of a ) = f CBD = (adjacent s in a right angle) CDB = (base s of an isosceles, CB = CD) DCE = (alternate s, CE BD) CED = (base s of an isosceles, DC = DE) = b LKM = ( K bisects JKM) KMN = (alternate s, KL NM) = d DIH = (corresponding s, DE FG) DEJ = (corresponding s, GH EJ) = f TUX = (adjacent s on a straight line) TUV = (sum of adjacent s) STU = (alternate s, ST UV) = h DBC = (adjacent s in a right angle) BDE = (co-interior s, DE BC) DEG = (alternate s, BD EG) DEF = ( s at a point) = a BCD = (alternate s, AB CE) BDE = (eterior of a ) = c ZXY = ( sum of a ) WXV = (verticall opposite s) VWX = (base s of isosceles, VW = VX) WVX = ( sum of a ) = e LMN = (corresponding s, JK LM) LNO = (eterior of a ) ONP = (OK bisects LNP) = g EDG = ( s at a point) EFG = ( sum of a quadrilateral) GFH = 9 (adjacent s on a straight line) = 9

7 Answers 9 h QRS = (alternate s, PQ RS) UTS = (opposite s of a parallelogram) STV = (adjacent s in a right angle) = j UWV = ( in an equilateral ) VWX = ( s at a point) WVY = (co-interior s, VY WX) = l IML = ( in a regular pentagon) MLN = (alternate s, LN IM) LMN = ( sum of a ) = b PQ SR (opposite sides of a rectangle are parallel) QPR = (alternate s, PQ SR) = a BC = CD (sides of a rhombus) CBD = (base s of an isosceles, BC = CD) m = c AD BC (opposite sides of a rhombus are parallel) ABC = (co-interior s, AD BC) EBC = (diagonals of a rhombus bisect s at the vertices) m = e BEC = 9 (diagonals of a rhombus are perp.) FEC = (alternate s, EF DC) BEF = 9 (adjacent s in a right angle) m = 9 i VWX = (co-interior s, UV XW) XWZ = ( s at a point) WXY = ( sum of a quadrilateral) = k OA = OB (equal radii) OAB = (corresponding s, AB CD) OBA = (base s of an isosceles, OA = OB) EOB = (eterior of a ) = a QRS = 9 ( in a rectangle) QSR = ( sum of a ) = c PT = TS (diagonals of a rectangle are equal and bisect each other) PST = (base s of an isosceles, PT = TS) PTS = ( sum of a ) QTR = (verticall opposite s) = b AEB = 9 (diagonals of a rhombus are perp.) ABE = ( sum of a ) CBE = (diagonals of a rhombus bisect s at the vertices) m = d ADC = (adjacent s on a st. line) ADB = (diagonals of a rhombus bisect s at the vertices) m = f DEF = (base s of an isosceles, DE = DF) EDF = ( sum of a ) AB DC (opposite sides of a rhombus are parallel) DAB = (co-interior s, AB DC) m = Eercise. Let AEC AED = α (adjacent s on a straight line) DEB = ( α) (adjacent s on a straight line) AEC = DEB (both equal to α) Let POR POS (PO bisects ROS) ROQ = α (adjacent s on a straight line) SOQ = α (adjacent s on a straight line) ROQ = SOQ (both equal to α) a Let ABD DBC = 9 α (adjacent s in a right angle) EBC (adjacent s in a right angle) ABD = EBC (both equal to α) b ABE + DBC = 9 + α + 9 α = Let ABD and FBC = β DBE (DB bisects ABE) EBF = β (FB bisects EBC) + β = ( s on a straight line) α + β = 9 DBF + β (sum of adjacent s) = 9 DB BF

8 9 Mathscape Etension Let TWX TUY (corresponding s, WX UY) YUV (UY bisects TUV) UVZ (alternate s, UY ZV) TWX = UVZ (both equal to α) Let AEF, CAE = β and ACD = γ BAE = α (co-interior s, AB EF) ( α) + β + γ = (co-interior s, AB CD) α = β + γ 9 Let ABP PBD (PQ bisects ABD) ABD (sum of adjacent s) BDE (alternate s, AB CE) BDS (RS bisects BDE) PBD = BDS (both equal to α) PQ RS (alternate s are equal) Let ABC and CDE = β Construct FC, parallel to BA and DE BCF (alternate s, BA FC) DCF = β (alternate s, FC DE) BCD + β (sum of adjacent s) BCD = ABC + CDE Let CHG GHD = α (adjacent s on a straight line) CDI = α (alternate s, GH DJ) DIE = ( α) (co-interior s, CD EF) JIF (verticall opposite s) CHG = JIF (both equal to α) Eercise. Construct DE, through B, parallel to AC DBA = BAC (alternate s, DE AC) EBC = BCA (alternate s, DE AC) DBA + ABC + EBC = ( s on a straight line) BAC + ABC + BCA = C = A + B (given) A + B + C = ( sum of a ) A + B + ( A + B) = ( A + B) = A + B = 9 C = 9 ABC is right-angled Let BAC BCA (base s of isosceles, AB = BC) DCE (verticall opposite s) DEC (alternate s, AB DE) CDE is isosceles ( DCE = DEC) Let PRS QRS (SR bisects PRQ) PRQ (sum of adjacent s) PQR (base s of isosceles, PQ = PR) PSR (eterior of QRS) PSR = PRS Construct CE, parallel to AB ECD = BAC (corresponding s, AB CE) BCE = ABC (alternate s, AB CE) BCD = ECD + BCE (sum of adjacent s) BCD = BAC + ABC Let BCA DAC = 9 α ( sum of ACD) EBC = 9 α ( sum of BCE) DAC = EBC (both equal to 9 α) Let CAB ACB (base s of isosceles, AB = BC) EBD (corresponding s, AC BE) CBE (alternate s, AC BE) CBE = EBD (both equal to α) EB bisects CBD BAD = (given) ABD = (base s of isosceles, AD = BD) DBC = (adjacent s in a right angle) BCD = ( sum of ABC) BDC = (eterior of ABD) BCD is equilateral (all s are )

9 Answers 9 9 Let YXW and XYW = β a XWZ + β (eterior of XYW) b WXZ (XW bisects YXZ) XZY = β (base s of isosceles, XY = XZ) XWY + β (eterior of XWZ) c XWY + XWZ = (adjacent s on a straight line) (α + β) + (α + β) = α + β = α + β = 9 XWZ = XWY = 9 XW YZ a OA = OB = OC (equal radii) b Let OAC and OBC = β OCA (base s of isosceles, OA = OC) OCB = β (base s of isosceles, OB = OC) α + β = ( sum of ABC) α + β = 9 ACB + β (sum of adjacent s) ACB = 9 Let ABG and CDG = β GBD (EB bisects ABD) GDB = β (FD bisects BDC) α + β = (co-interior s, AB CD) α + β = 9 BGD = (α + β) ( sum of a ) = 9 = 9 EB FD a BDC + β (eterior of ABD) b BCE = BAD (given) DBC = β (BD bisects ABC) DEC + β (eterior of BCE) BDC = DEC (both equal to α + β) CD = CE (equal sides lie opposite equal s) Let ABF and ACE = β EBC (EB bisects ABC) ACB (base s of isosceles, AB = AC) BEC (alternate s, AB EC) ECD = β (EC bisects ACD) α + β = (adjacent s on a straight line) α + β = 9 BFC + β (eterior of CEF) BFC = 9 AC BE a Let BAD ABD (base s of isosceles, AD = DB) DBC = 9 α (adjacent s in a right angle) ACB = 9 α ( sum of ABC) DBC = ACB (both equal to 9 α) BCD is isosceles b AD = DB (given) DB = DC (equal sides lie opposite equal s, DBC = ACB) AD = DC (both equal to DB) D is the midpoint of AC a Let ABE and BAC = β BEC + β (eterior of ABE) EBC (BE bisects ABC) BCD = (α + β) + α + β (eterior of BCD) b BAC + BCD = β + α + β = (α + β) = BEC Let UWY WUX = 9 α ( sum of UWX) VYW (base s of isosceles, VW = VY) XZY = 9 α ( sum of XYZ) UZV = 9 α (verticall opposite s) WUX = UZV (both equal to 9 α) UVZ is isosceles Let ABG and CDE = β GBC (GB bisects ABC) ABC (sum of adjacent s) BCF = α (co-interior s, AB FE) CED = β (base s of isosceles, CD = CE) ECD = β ( sum of CDE) BCF = ECD (verticall opposite s) α = β α = β GBC = CDE GB DE (alternate s are equal)

10 9 Mathscape Etension Eercise. a AAS, b RHS, c SSS, d SAS a es, SAS, b no, c es, RHS, d no no, AAA is not a congruence test a QR = SR (given) PR = TR (given) QRP = SRT (verticall opposite s) PQR TSR (SAS) b DF = FH (given) DEF = FGH (alternate s, DE GH) DFE = GFH (verticall opposite s) DEF HGF (AAS) c XY = XW (given) YZ = WZ (given) XZ is a common side XYZ XWZ (SSS) a AE = BE (CD bisects AB) CE = DE (AB bisects CD) AEC = BED (verticall opposite s) ACE BDE (SAS) AC = BD (matching sides of congruent s) c WXY = WZY (given) XYW = ZYW (WY bisects XYZ) WY is a common side XYW ZYW (AAS) WX = WZ (matching sides of congruent s) XWZ is isosceles e QPR = SRT (corresponding s, PQ RS) QRP = STR (corresponding s, QR ST) PR = RT (QR bisects PT) PQR RST (AAS) PQ = RS (matching sides of congruent s) g CD = DE (given) DF is a common side DFC = DFE = 9 (DF CE) CDF EDF (RHS) CF = FE (matching sides of congruent s) DF bisects CE i AB = BD (BC bisects AD) BC = DE (given) ABC = BDE (corresponding s, BC DE) ABC BDE (SAS) BAC = DBE (matching s of congruent s) AC BE (corresponding s are equal) a PQ = PR (given) QX = RY (given) PQX = PRY (base s of isosceles, PQ = PR) PQX PRY (SAS) b PX = PY (matching sides of congruent s) PXY is isosceles a ABC = ACB (base s of isosceles, AB = AC) BLC = CMB = 9 (CL AB, BM AC) BC is a common side BLC CMB (AAS) c LN = MN (matching sides of congruent s) d LM = MN (given) MK is a common side MKL = MKN = 9 (MK LN) MLK MNK (RHS) b OI = OK (equal radii) OJ is a common side OJI = OJK = 9 (OJ IK) OIJ OKJ (RHS) IOJ = KOJ (matching s of congruent s) OJ bisects IOK d OA = OD (equal radii) OB =OC (equal radii) AB = CD (given) AOB DOC (SSS) AOB = COD (matching s of congruent s) f IJL = JLK (alternate s, IJ LK) ILJ = LJK (alternate s, LI KJ) LJ is a common side IJL KLJ (AAS) IJL = KJL (matching s of congruent s) LJ bisects IJK h RS = QT (given) RT is a common side SRT = RTQ (alternate s, RS QT) RST TQR (SAS) RTS = QRT (matching s of congruent s) QR TS (alternate s are equal) j OX = OY (equal radii) XM = MY (OM bisects XY) OM is a common side OMX OMY (SSS) OMX = OMY (matching s of congruent s) But, OMX + OMY = (adjacent s on a straight line) OMX = OMY = 9 OM XY a AB = AC (given) PB = QC (P, Q are midpoints of equal sides) b PB = QC (proven above) BC is a common side ABC = ACB (base s of isosceles, AB = AC) PBC QCB (SAS) PC = QB (matching sides of congruent s) b BL = CM (matching sides of congruent s) BLN = CMN = 9 (CL AB, BM AC) BNL = CNM (verticall opposite s) BLN CMN (AAS)

11 Answers 9 9 a Construct AD, the angle bisector of A AB = AC (given) BAD = CAD (AD bisects BAC) AD is a common side BAD CAD (SAS) ABC = ACB (matching s of congruent s) a AC = CB (sides of an equilateral ) ACD = BCD (CD bisects ACB) CD is a common side ACD BCD (SAS) A = B (matching s of congruent s) b AB = AC (sides of an equilateral ) BAE = CAE (AE bisects BAC) AE is a common side ABE ACE (SAS) B = C (matching s of congruent s) c A = B (proven above) B = C (proven above) A = B = C But, A + B + C = ( sum of a ) A = B = C = QM = PN (given) PM = SN (given) LMN = LNM (base s of isosceles, LM = LN) QMP PNS (SAS) PQ = PS (matching sides of congruent s) PQS = PSQ (base s of isosceles, PQ = PS) b Construct AD, the angle bisector of A ABC = ACB (given) BAD = CAD (AD bisects BAC) AD is a common side ABD ACD (AAS) AB = AC (matching sides of congruent s) i Let ABC and PBC = β ABP β (b subtraction) ACB (base s of isosceles, AB = AC) PCB = β (base s of isosceles, PB = PC) ACP β (b subtraction) ABP = ACP (both equal to α β) ii AB = AC (given) ABP = ACP (proven above) PB = PC (given) ABP ACP (SAS) BAP = CAP (matching s of congruent s) PA bisects ABC i BAC = ( in an equilateral ) DAC = (adjacent s on a straight line) ACB = ( in an equilateral ) BCE = (adjacent s on a straight line) ACD = (adjacent s in a right angle) ABC = ( in an equilateral ) CBE = (adjacent s in a right angle) ii AC = BC (sides of an equilateral ) ACD = CBE = (proven above) DAC = BCE = (proven above) ACD CBE (AAS) CE = AD (matching sides of congruent s) Eercise. BAC + ABC + BCA = ( sum of ABC) DAC + ADC + DCA = ( sum of ADC) ( BAC + DAC) + ABC + ( BCA + DCA) + ADC = (b addition) A + B + C + D = a ABP = PDC (alternate s, AB DC) BAP = PCD (alternate s, AB DC) AB = DC (opp. sides of a parallelogram) APB CPD (AAS) b AP = PC (matching sides of congruent s) BP = PD (matching sides of congruent s) c the diagonals of a parallelogram bisect each other. a BAC = ACD (alternate s, AB DC) BCA = CAD (alternate s, BC AD) AC is a common side ABC CDA (AAS) b AB = DC (matching sides of congruent s) AD = BC (matching sides of congruent s) c ABC = ADC (matching s of congruent s) d the opposite sides of a parallelogram are equal, and the opposite angles of a parallelogram are equal. a BCA (base s of isosceles, AB = BC) DAC (alternate s, AD BC) b BDC = β (alternate s, AB DC) BC = CD (sides of a rhombus are equal) CBD = β (base s of isosceles, BC = CD) c the diagonals of a rhombus bisect the angles at the vertices.

12 9 Mathscape Etension a BCA (base s of isosceles, AB = BC) CBD = β (diagonal of a rhombus bisects the at the verte) b α + β = ( sum of ABC) α + β = 9 c BPC + β (eterior of ABP) BPC = 9 AC BD d the diagonals of a rhombus are perpendicular. a i α + β = ( sum of a quadrilateral) α + β = ii BAD + ADC + β = AB DC (co-interior s are supplementar) BAD + ABC + β = AD BC (co-interior s are supplementar) bi AB = CD (given) BC = AD (given) AC is a common side ABC CDA (SSS) ii BAC = ACD (matching s of congruent s) AB CD (alternate s are equal) BCA = CAD (matching s of congruent s) BC AD (alternate s are equal) c i AB = DC (given) BAC = ACD (alternate s, AB DC) AC is a common side ABC CDA (SAS) ii BCA = CAD (matching s of congruent s) AD BC (alternate s are equal) di AP = PC (BD bisects AC) BP = PD (AC bisects BD) APB = CPD (verticall opposite s) APB CPD (SAS) ii AB = DC (matching sides of congruent s) ABP = PDC (matching s of congruent s) AB DC (alternate s are equal) a ADP (base s of isosceles, PA = PD) ABP = β (base s of isosceles, PA = PB) b α + β = ( sum of ABD) α + β = 9 DAB + β (sum of adjacent s) DAB = 9 c ABCD is a parallelogram (diagonals bisect each other) But, DAB = 9 (proven above) ABCD is a rectangle (parallelogram with one angle a right angle). a AB = CD (opposite sides of a rectangle are equal) ABC = BCD = 9 (angles in a rectangle) BC is a common side ABC DCB (SAS) b AC = BD (matching sides of congruent s) c the diagonals of a rectangle are equal. a AP = PC (BD bisects AC) BPA = BPC = 9 (BD AC) BP is a common side ABP CBP (SAS) b AB = BC (matching sides of congruent s) c ABCD is a parallelogram (diagonals bisect each other) AB = BC (proven above) ABCD is a rhombus (parallelogram with a pair of adjacent sides equal) 9 i α = ( sum of a quadrilateral) α = 9 ii BAD + ADC + α = AB DC (co-interior s are supplementar) BAD + ABC + α = AD BC (co-interior s are supplementar) ABCD is a parallelogram (two pairs of opposite sides parallel) But, ABC = 9 ABCD is a rectangle (a parallelogram with one angle a right angle). a PBR = RDC (alternate s, AB DC) BPR = RQD (alternate s, AB DC) BR = RD (AC bisects BD) BPR DQR (AAS) b PB = DQ (matching sides of congruent s) AB = DC (opposite sides of a parallelogram are equal) AB PB = DC DQ AP = QC

13 Answers 9 a BDC (alternate s, AB DC) ADB (BD bisects ADC) AB = AD (equal sides lie opposite equal s) b ABCD is a rhombus (parallelogram with a pair of adjacent sides equal). a Let DBC ADB (alternate s, AB DC) FBC = α (adjacent s on a st. line) ADE = α (adjacent s on a st. line) FBC = ADE (both equal to α) b FBC = ADE (proven above) BC = AD (opposite sides of a parallelogram are equal) BF = ED (given) FBC EDA (SAS) c BFC = DEA (matching s of congruent s) FC AE (alternate s are equal) FC = AE (matching sides of congruent s) AFCE is a parallelogram (one pair of opposite sides equal and parallel). Eercise. a = 9, b a =, c t =, d m =, e u =, f e = a no, b es, c no, d es abd = cm, BC = cm, b AC = AB + BC ; ABC is right-angled. cm m cm = cm, cm, cm cm b AC = +, BC = +, c AB = AC + BC a AB = AD + BD, b CD = BC BD d In a quadrilateral in which the diagonals are perpendicular, the sum of the squares on the opposite sides are equal. Chapter Review a =, b p =, c a =, d c =, e e =, f s =, g m =, h b = a ABD + DBC = 9, AB BC, b PQS + SQR =, P, Q, R are collinear. a no, corresponding s are not equal, b es, co-interior s are supplementar, c es, alternate s are equal a, b a,, b,, c,, d,, e,, f, a, b, c a, b, c no 9 a Let ABG CDG = ABG (corresponding s, AB CD) DEF = ABG (given) CDE = DEF (alternate s, CD EF) Now, CDE = CDG (both equal to α) CD bisects GDE c Let IMH IMN = 9 + α (b addition) JIM = (9 + α) (co-interior s, IJ MN) = 9 α JIK = JIM (IM bisects JIK) = α IKL (co-interior s, IJ KL) IKL = IMH e Let GBD ABG = GBD (BG bisects ABD) BDF = 9 α ( sum of BED) CDF = BDF (FD bisects BDC) = 9 α Now, ABD + BDC + (9 α) = AB CD (co-interior s are supplementar) b Let QSR PQS = α (co-interior s, PQ RS) QST = 9 α (TS RS) TSU = 9 (9 α) (QS SU) Now, PQS + TSU =, PQS and TSU are supplementar. d Let PQS SPQ = PQS (base s of isosceles, PS = SQ) SQR = 9 α (PQ QR) PRQ = 9 α ( sum of PQR) Now, SQR = PRQ (both equal to 9 α) QRS is isosceles f Let EAC and ECA = β ABC = EAC (given) BCD = ECA (DC bisects ACB) = β AED + β (et. of ACE) ADE + β (et. of BCD) Now, AED = ADE (both equal to α + β) ADE is isosceles

14 9 Mathscape Etension a i AB = BC (given) OA = OC (equal radii) OB is a common side OAB OCB (SSS) ii AOB = COB (matching s of congruent s) OB bisects AOC b i PQS = QSR = 9 (alternate s, PQ SR) PS = QR (given) QS is a common side PQS RSQ (RHS) ii PSQ = RQS (matching s of congruent s) PS QR (alternate s are equal) PQ SR (given) PQRS is a parallelogram (opposite sides are parallel) c i XWZ = XYZ (given) XZW = XZY (XZ bisects WZY) XZ is a common side XWZ XYZ (AAS) ii XW = XY (matching sides of congruent s) WXY is isosceles a PQ = SR (opposite sides of a parallelogram) PQT = RST (alternate s, PQ SR) PTQ = STR (vert. opp. s) PQT RST (AAS) b PT = TR (matching sides of congruent s) QT = TS (matching sides of congruent s) the diagonals bisect each other d i CD = EF (given) DCE = FEG (corresponding s, CD EF) CE = EG (EF bisects CG) CDE EFG (SAS) ii DE = FG (matching sides of congruent s) a KN = LM (opposite sides of a rectangle) KNM = LMN = 9 ( s in a rectangle) NM is a common side KNM LMN (SAS) b KM = LN (matching sides of congruent s) diagonals of a rectangle are equal a XY = YZ (sides of a rhombus) a diagonals bisect each other XZY = ZXY (base s of isosceles, XY = YZ) b SW is a common side SWV = SWT = 9 (SU VT) WXZ = XZY (alternate s, WX ZY) VW = WT (SU bisects VT) SVW STW (SAS) Now, WXZ = ZXY (both equal to α) c SV = ST (matching sides of congruent s) ZX bisects WXY STUV is a rhombus (parallelogram with a pair of b WX = XY (sides of a rhombus) adjacent sides equal) XWY = XYW (base s of isosceles, WX = XY) = β α + β = ( sum of WXY) α + β = 9 WAX + β (eterior of XAY) = 9 XZ WY a AB = AC (given) BR = CS (R, S are midpoints of AB, AC) b BR = CS (proven above) ABC = ACB (base s of isosceles, AB = AC) BC is a common side RBC SCB (SAS) c CR = BS (matching sides of congruent s) d TR = TS (given) CR TR = BS TS TB = TC BTC is isosceles AC = AB + BC AD = AC + CD = AB + (AB) = AB + (AB) = AB = 9AB AD = AB MP LP a = (given) LP PN a LP = LP b LP = ab LP = ab b LM = a + ab, LN = b + ab c LM + LN = a + ab + b + ab = a + ab + b = (a + b) = MN LMN is right-angled (converse of Pthagoras theorem)

15 Answers 9 Factorisation and algebraic fractions Eercise. + + a + + +, b pq p + q, c uv u v + a + +, b m + m +, c u + u, d b + b, e a a, f t t +, g c c +, h z z +, i d + d, j + +, k m + m +, l a a, m g g +, n t t, o n + n, p r + r +, q k + k, r v v + a + +, b p + p +, c a + a +, d t 9t +, e b b +, f c c +, g z + z, h d d, i s + s, j e + e, k u + u, l k k, m f + f +, n w w, o r r +, p g + g, q h + h, r v + v +, s q q +, t m + m, u i 9i, v l + l, w 9, j j + a 9a aa + ab + b, b m + mn + n, c g gh + h, d p pq + q aa + a + 9, b p p +, c c + c +, d t t +, e u u +, f k + k +, g s + s + 9, h p p + 9a + + 9, b 9t t +, c m + m +, d c c +, e 9 g + g, f + r + r, g 9p pq + q, h 9a + ab + b, i e ef + f, j c + cd + d, k 9g gh + 9h, l j + jk + k, m p q + pqr + r, n a b abcd + 9c d, o a no, b es, c no, d es, e es, f no, g no, h no, i es a, b p, c +,, d a, a, e k +,, f u,, g b, h e +, e, i n +,, j z, 9 a m n, b p, c r, d 9 g, e 9, f w, g t, h k, i b, j e 9, k r, l 9c, m 9h, n n, o p q, p 9z, q 9s t, r a b c a a + 9a +, b t, c p p +, d e e +, e a + a +, f + +, g m, h + 9, i k k +, j c, k h +, l m a + + +, b a a + a, c k + k + k +, d n n n +, e p + p + p + 9, f + 9, g t + t + t +, h e 9e + e Eercise. a (c + ), b (m + ), c ( + e), d ( + ), e (g ), f (k ), g (9 r), h ( t), i ( + ), j (m n), k (p + q), l (f g), m ( + z), n b(a c), o m(m + ), p c(c ) a (n + ), b (b + ), c ( ), d (u ), e (p + ), f (g ), g (w + ), h (z ), i (h ), j (d + ), k (q ), l (f ), m ( 9k), n ( + 9v), o ( a), p ( s) a ( + z), b p(q + r), c g(f h), d c(d e), e j(i + k), f n(m p), g v(u w), h t(s + u), i b(b + ), j a(a ), k q(q ), l u( + u), m pq(r + s), n de(c f), o ( + ), p ab(c b), q fg(g fh), r k(j + km), s tu(u + tv), t gh(gh i) a (p + q + r), b a(b + c d), c ( + z), d (e + f + g), e (m m + n), f ( v v ), g (c c + ), h a(b + b ), i r(r s ), j ( z + ), k i(j + i k), l z( z), m pq(p + + q), n rs(9 s r), o abc(a + b c) a (k + ), b (n + ), c (c + ), d (w + ), e ( ), f 9(d ), g (m ), h (g ), i 9( + ), j ( e), k ( + z), l ( 9t), m d(c + e), n j(i k), o a(a + ), p v( v), q n(n ), r b( + b), s f( 9e), t cd(c + d) am ( + m), b ( + ), c t ( t ), d ( ), e a ( + a ), f g ( g ), g u (u + ), h h ( h), i c (c + ), j g (g + ), k q ( q ), l z (9z ) Eercise. a 9, ( + )( ), b, ( + )( ) a (p q)(p + q), b (c d)(c + d), c (m n)(m + n), d (u v)(u + v) a ( )( + ), b (a )(a + ), c (p )(p + ), d ( )( + ), e (z )(z + ), f (c )(c + ), g (t )(t + ), h (b 9)(b + 9), i ( k)( + k), j ( g)( + g), k ( m)( + m), l ( u)( + u) a (e )(e + ), b (h )(h + ), c ( s)( + s), d (9 j)(9 + j) a (a )(a + ), b (p )(p + ), c (q )(q + ), d (c )(c + ), e ( )( + ), f ( r)( + r), g ( u)( + u), h ( 9t)( + 9t), i (a b)(a + b), j ( )( + ), k (e f)(e + f), l (j k)(j + k), m (g h)(g + h), n (m n)(m + n), o (p q)(p + q), p (s t)(s + t), q (ab c)(ab + c), r (p qr)(p + qr), s ( z)( + z), t (ef 9gh)(ef + 9gh) a (m )(m + ), b (a )(a + ), c (t )(t + ), d ( )( + ), e ( )( + ), f ( p)( + p), g ( e)( + e), h ( z)( + z), i (n )(n + ), j (c )(c + ), k ( f)( + f), l (k )(k + ), m a(a )(a + ), n n ( n)( + n), o d(d )(d + ), p u( u)( + u), q h(h )(h + ), r w(w )(w + ), s s( s)( + s), t j(j )(j + ) a (a )(a + ), no each factor still has a common factor of. b (a )(a + ) a 9(k )(k + ), b (c )(c + ), c ( )( + ), d (e f)(e + f) 9 a, b 9, c a (a + b c)(a + b + c), b (m n p)(m n + p), c ( + )( + + ), d (j k )(j k + ), e (b )(b + ), f p(p + ), g (m + n )(m + n + ), h (c d )(c d + ), i (p + q r)(p + q + r)

16 9 Mathscape Etension Eercise. a (c + d)(a + b), b ( + )( + ), c (p )(n + ), d ( + )(w z), e (t )(t 9), f (a )(g h), g (q + r)(p + s), h (a + )( c), i ( + )(mn ), j (d + e)(u + ), k (p q)( + w), l (i j)(h ), a (m + n)(k + ), b (c + d)(a + b), c (p + q)(p + ), d (w + )( + ), e (e )(c + d), f (h )(g + ), g (v + )(u + ), h ( + )( + a), i (n )(m + p), j (k + h)(g + ), k (r + )(pq + ), l (m + p)(n + ), m (a b)( + c), n (e + f)( + e), o (a + )(a + ) a (c + d)( e), b (q + s)(p r), c (p q)(p ), d ( )( ), e (h + i)(g ), f (u v)( w), g ( )( ), h (k + )(k m), i (m )(jk ), j (p q)(n ), k (z w)(z u), l ( + )( z) a ( + z)( + w), b (p + q)(r + ), c (n + p)(m + k), d ( + )(z + ), e (d + )(c + ), f (e + )(f + ) a ( )( ), b (z )( z), c (b c)(a + b), d (a b)(a b), e ( n)(m + n), f (m n)( + p), g (v w)(u + w), h (d c)(e ), i ( )(w ), j (c d)(c + d), k (q r)(p q), l (s t)(9r + s) a ( + )( + + ), b (m )(m + n), c (k )(k + m), d (a + b)(a + b + c), e ( )( w), f (g + h)(f + g + h) a (c d)(c + d + ), b (p q)(p q + r) Eercise. a,, b,, c,, d,, e,, f,, g,, h, 9, i,, j,, k,, l 9, bia >, b >, ii a <, b <, iii a >, b <, iv a <, b > a ( + )( + ), b ( + )( + ), c (u + )(u + ), d (m + )(m + ), e (a + )(a + ), f (t + )(t + ), g (k + )(k + ), h (p + )(p + ), i (n + )(n + 9), j (d + )(d + ), k (s + )(s + ), l (b + )(b + 9), m (e + )(e + ), n (c + )(c + ), o (r + )(r + ), p (z + )(z + ) a (m )(m ), b (q )(q ), c (d )(d ), d (a )(a ), e (u )(u 9), f (e )(e ), g (n )(n ), h (w )(w ), i (h )(h 9), j (v )(v ), k (t )(t ), l (s )(s ), m (k )(k ), n (j )(j ), o ( )( 9), p (f )(f ) a ( + )( ), b (d + )(d ), c (a + )(a ), d (p + )(p ), e (v )(v + ), f (u 9)(u + ), g (m )(m + ), h ( )( + ), i (f + )(f ), j (w 9)(w + ), k (k 9)(k + ), l (c + )(c ), m (z )(z + ), n (i + )(i ), o (r )(r + 9), p (e )(e + ), q (s + )(s ), r (h + )(h ), s (b )(b + ), t (t + )(t 9) a (n )(n ), b (c + )(c + ), c ( )( + ), d (d + )(d ), e (q + )(q + ), f (t )(t ), g (v + )(v + ), h (j )(j + ), i (g )(g + ), j (b + )(b + ), k (r 9)(r + ), l (u )(u ), m (e + )(e + ), n (l )(l + ), o ( )( ), p (p + )(p ), q (z )(z + ), r (a + )(a ), s (f )(f ), t (m + )(m + ), u (w 9)(w + ), v (k + )(k + ), w (h )(h ), (i + )(i ) a (p + ), b (c + ), c (g ), d ( ), e (t + ), f (r ), g ( 9), h (j + ) a (m )(m ), b (k + )(k + ), c (a + )(a ), d (c 9)(c + ), e (t )(t ), f ( )( + ), g (d )(d ), h (n + )(n ), i ( )( + ) 9a +, b g + a ( )( + )( )( + ), b ( )( +)( )( + ), c ( )( )( + ), d ( +)( )( + ), e ( + )( )( + ), f ( )( + )( )( + ), g ( )( )( + ), h ( )( + )( )( + ), i ( )( + )( )( + ), j ( )( )( + ), k ( + )( )( + ), l ( )( + )( )( + ) Eercise. B C D A a ( + )( + ), b ( + )( + ), c ( + )( + ), d ( + )( + ), e ( + )( + ), f ( + )( + ), g ( + 9)( + ), h ( + )( + ), i ( + )( + ), j ( )( ), k ( )( ), l ( )( ), m ( )( ), n ( )( ), o ( )( ), p ( )( ), q ( )( ), r ( )( ) a ( )( + ), b ( + )( ), c ( )( + ), d ( )( + ), e ( + )( ), f ( + )( ), g ( + 9)( ), h ( + )( ), i ( )( + ), j ( + )( ), k ( )( + ), l ( + )( ), m ( )( + ), n ( )( + ), o ( )( + ), a (k + )(k + ), b (c + 9)(c ), c (n 9)(n ), d ( + )( ), e (p )(p ), f (a + )(a ), g (b + )(b + ), h (u )(u + ), i (w + )(w ), j (h + )(h + ), k (j )(j + ), l (l )(l 9) a (a + )(a + ), b (n )(n ), c (k + 9)(k ), d (p )(p + ), e (c + )(c + ), f (e )(e + 9), g (t )(t ), h (b )(b + 9), i (m )(m ), j ( + )( + ), k (9w + )(w ), l (q )(q + ) 9a ( )( + ), b ( a)( + a), c ( p)( + p), d ( m)( m), e ( + g)( + g) f ( w)( + w) a (k + )(k + ), b (p )(p ), c (a + )(a ), d (v + )(v ), e (f )(f ), f (e )(e + ) a a +, b n + 9 a ( + )( + ), b ( )( ), c ( )( + ), d ( + )( ), e ( )( ), f ( )( + ) Eercise. a (e + ), b (a )(a + ), c (m + )(m + ), d (n + )(n + p), e ( + )( + ), f ( k)( + k), g ( ), h (t )(t ), i (w + ), j (p )(a + ), k ( + ), l (q )(q ), m (g h)(g + h), n (j k), o (u )(u + ), p (b + )(b + ), q t(t 9u), r (c )(c + ), s ( )( + 9), t fg(e + h), u (h + )(h ), v (v w)(v + w), w (g )(m ), (f )(f + ) a 9p(q + p), b ( )( + ), c (e f)(e + f), d (b + )(b ), e (g + h)( i), f ( ), g (z + )(z ), h (pq r)(pq + r), i ( k)( + k), j ( kl)( + kl), k (m + )(m n), l (s )(s 9), m 9(h )(h + ), n ( + )( ), o ( j)( k),

17 Answers 99 p n(n )(n + ), q (a + b)(a + b + c), r ( + t)( + t ), s (u + ), t (f )(f + ), u (a b)( + c), v a(a + b ), w k(k + 9)(k ), v(u + )(u ) a ( + )( + + z), b (t + )(t )(t + ), c (w )(w + )(w )(w + ), d b (a c)(a + c), e ( + n )( n)( + n), f ( + )( ), g (a + b + c + d)(a + b c d), h (a )(a + )(a b), i ( )( + )(g + ) Eercise. a t k m c a q p b g e a --, b --, c -----, d , e -----, f -----, g --, h -----, i --, j --, k --, l --, m -----, n -----, o -----, p u, q --, d g c s g s v r d a t r ---, s -----, t a +, b k, c m +, d , e , f b + c, g , h +, i , w c v w t c d n + m g h a j , k , l , m --, n --, o --, p -- am +, b , c , d , e t +, p g z + n p q w + c e + z 9 s f h +, g , h a +, b a, c , d , e , f , g , h , ( k ) p+ q n c e z s + d + p + k + h + 9 v + k + u + i a +, b , c , d , e , f ap + q, b , c , ( d ) r k + h v m + v w + b c n p d , e , f a, b, c, d, e a, f, g, h, i ( a + ) n c 9 s g ( + k) p k + g Eercise.9 9c eq mn s d ( b ) ( ) a -----, b --, c -----, d , e , f -----, g ----, h a, b --, c , d --, e , 9 cp 9t 9c 9 ( k + ) ac ca ( + b) f a --, b --, c , d u m a + s t , e , f a , b --, c , d , ( k + ) c 9 t n ( t ) c + n + r + a+ b bb ( ) ( w + ) e --, f , g , h , i , j , k a + e + c+ d , l , m , n k c + n r ab w a e d n + ( a ) z + r m + q a --, b , c --, d , e , f --, g --, h , i , j , k , l -----, m, k n + z r m q + v c + n n a -----, b --, c , d ---- c 9 z q Eercise. z a m k 9 f + 9n 9t + b + a ----, b -----, c ---, d --, e -----, f , g -----, h a , b , c , d , 9u k + z e , f a , b , c , d , e , ( + ) ( + ) ( ) ( + ) ( + ) ( + ) ( + ) ( + ) f , g , h , i , j , k , l ( ) ( + ) ( + ) ( + ) ( ) ( + ) ( + ) ( + ) ( + ) ( ) ( ) a , b , c a --, b , c , ( + ) ( + ) ( + ) ( ) ( + ) ( + ) ( + ) ( + ) ( + ) ( + ) ( + ) d , e , f , g ( + ) ( ) ( + ) ( ) ( + ) ( ) ( + ) ( + ) ( + ) ( + ) ( + ) 9 + h , i , j , k ( ) ( + ) ( + ) ( ) ( ) ( + ) ( ) ( + ) ( ) ( ) ( + ) l a , b , c , d , ( ) ( + ) ( + ) ( + ) ( ) ( + ) ( ) ( + ) e , f , g , h , ( ) ( + ) ( + ) ( ) ( ) ( + ) ( + ) ( + ) ( + ) ( 9) ( + ) ( + 9)

18 Mathscape Etension i , j , k , l ( + ) ( ) ( ) ( ) ( + ) ( ) ( ) ( + ) ( + ) ( + ) ( + ) ( ) 9 a , b , c , d , e , ( ) ( + ) ( ) ( + ) ( ) ( + ) ( ) ( + ) f , g , h , i , ( ) ( + ) ( ) ( ) ( ) ( ) ( ) ( + ) ( ) ( + ) ( ) j , k , l ( ) ( + ) ( 9) ( ) ( + ) ( + ) ( ) ( + ) ( ) Chapter Review a + 9 +, b m m +, c t + t, d a a a p p +, b r r am m + 9, b c + c +, + a (t + ) = t + t +, b (n 9) = n n +, c ( + ) = 9 + +, d (u ) = u u + a no, b no, c es, d no, e no, f es aa + a + 9a +, b n n +, c a 9, d u u a ( + ), b a(a + ), c (m + ), d (p q), e s(r + t), f pq(r s), g 9( ), h gh(g h), i a(b ), j c(9c + d) 9a (m n)(m + n), b (z )(z + ), c ( p)( + p), d ( r)( + r), e (w )(w + ), f (a )(a + ), g ( )( + ), h (9u v)(9u + v), i (ab c)(ab + c), j (pq rs)(pq + rs) a (z + )( + ), b (q r)(p ), c (e + f)(e ), d (k )(j ), e ( q)(p q), f ( d)(c + d) a ( + )( + ), b (b )(b ), c (e )(e + ), d (p + )(p ), e (a + ), f (q 9) a (t + )(t + ), b (m )(m ), c (c )(c + ), d (b + )(b ), e (s + )(s + ), f (d 9)(d + ) a (n )(n + ), b (v + )(v + ), c (k + ), d (e + )(e + ), e (a + )(a + b), f (h )(h + ), g (p )(p + ), h ( + z), i ( + u)( + u ), j (a b)(a + b), k (m + )(m ), l ( )( + ), m (d + e)(c ), n rs(s r + ), o (g + )(g ) a ( )( + ), b (a + )(a + ), c n(n )(n + ), d (h )(h + ), e a(b )(b ), f (u + )(u + ), g (z )(z + ), h ( )( ) t a p a --, b -----, c -----, d -- d v q h + k + p b c a e +, b , c , d --, e u +, f , g , h , i , j , k, a k + p a c a c a + h + l a , b , c --, d a , b , c , d , a c + ( + ) ( ) ( + ) e a , b , c , ( + ) ( + ) ( + ) ( ) ( ) ( + ) ( + ) ( ) ( 9) ( ) d ( ) ( + ) ( + ) Quadratic equations Eercise. ai, ii, iii, bi, ii, iii., ci, ii, iii --, di, ii, iii, ei, ii, iii -----, fi 9, ii 9 --, iii a no, b es, c es, d no, e es, f es, g es, h no, i es, j no, k no, l es, m no, 9 n es, o es, p no a = +, b =, c =, d =, e = +, f = a = + +, b = + +, c = +, d = +, e = +, f =, g = + +, h = + a =, b =, c =, d =, e =, f = a =,, b =,, c =, a =,, b =,, c =, 9 a ( + ) cm, b =, c cm cm Eercise. a =,, b a =,, c t =,, d m =,, e k =,, f c =,, g =,, h e =,, i p =,, j n =,, k z =,, l q =,, m w =, --, n f =, --, o u =, --, p a = --,, q q = --,, r c = --,, 9 s t = --, --, t v = --, --, u b = --, -----, v =, w = 9, m = -- aa =,, b =,, c p =,, d q =,, e e =,, f g =,, g n =,, h d =,, i r =,, j h = 9, 9, k f =,, l t =,, m k =,, n m =,, o =,, p =,, q a =,, r z = 9,, s n =,, t j =,, u r =, 9,

19 Answers v p =, w h =, v = a =,, b c =,, c d =, 9, d k =, --, e e =, --, f m =, -----, g =, h u = -----, i t = -- a = --,, b a = --,, c p = --,, d u = --,, e = --,, f c = --,, g e = --,, h k = --,, i m = --, 9, j v = --,, k g = --,, l d = --,, m t = --, --, n h = --, --, o n = --,, p w = --, --, q u = --, --, r s = --, --, s a = --, --, t j = --, --, u z = --, -- aa =,, b p =,, c s =,, d q =,, e b =,, f u =,, g =,, h u =,, i =,, j n =,, k p =, 9, 9 l r =,, m c =,, n u =,, o m =,, p z = --,, q w = --,, r k = --, a + =, b + =, c + + =, d + + =, e =, f + =, g =, h + 9 = Eercise. 9 a 9, +, b, +, c,, d,, e 9, +, f,, g,, h, + a --, + --, 9 9 b -----, + --, c -----, --, d --, --, e , , f -----, -- a =,, b =,, c =,, d =,, e =,, f =,, g =,, h =, 9, i =, a = ±, b = ±, c = ±, d = ±, e = ±, f = ±, g = ±, h = ±, i = ±, j = ±, k = ±, l = ± a =.,., b =.,., c =.,., d =.,., e =.9,., f =.,., g =.,., h =.,., i =.,., j =.,., k =.,., l =.,., m =.,.9, n =.,., o = 9.,., p =.,., q =.,., r =.,. a -----, + --, b -----, --, c , , d --, + --, e -----, --, f -----, --, g -----, + --, h -----, + --, i , -----, j -----, a =.,., b =.,., c =.,., d =.,.9, e =.9,., f =.,. Eercise. a =,, b =,, c =,, d =,, e =,, f =,, g =,, h =,, i =,, j =,, k =,, l =, a = --,, b = --,, c = --,, d = --,, e = --,, ± ± f = --,, g = --,, h = --,, i = --, a = , b = , ± ± 9 ± ± 9 ± ± c = , d = , e = , f = , g = , h = , ± ± ± ± i = , j = , k = , l = a = ±, b = ±, c = ±, d = ±, e = ±, f = ±, g = ±, h = ±, ± ± ± ± ± i = ±, j = , k = , l = , m = , n = , ± ± ± ± o = , p = , q = , r = a = --, b, c It is a perfect square. a There are no solutions. b b ac if the equation has solutions. a + =, b + 9 =, c = Eercise. a =,, b =,, c =,, d =,, e =.,., f =.,., g =,, h =,, i =,, j =.,., k =.,., l =.,., m = --,, n = --,, o = --,, --

20 Mathscape Etension p =.,., q =.,., r =.,. a =,, b =,, c =,, d =,, e =,, f =,, g = --,, h = --,, i = --,, j =.,., k =.,., l =.,.9 9 a =,, b =,, c =,, d =,, e = --,, f = --,, g =.,., h =.,., i =.9,.9, j =.9,.9, k =.,.9, l =.,. a =,, b =,, c =, 9, d = --,, e =,, f =,, g =.,., h =.,., i =.,., j = 9.,., k =.,., l =.9,. a =,, b =,, c =,, d =,, e =,, f = --,, ± ± ± ± g = , h = , i = , j = ±, k = ±, l = a = --,, b = --,, c = --,, d =.,., e =.,., f =.,. -- a =, = and =, =, b =, = and =, = 9, c =, = 9 and =, =, d =, = and =, =, e =, = and =, =, f =, = and =, =, g =, =, h =, = and =, =, i =, = and =, =, j =, = and =, =, k = --, = -- and =, =, l = --, = -- and =, = a =,,,, b =,,,, c =,,,, d =,, e =,, f =,, g =,,,, h =,, i =,,,, j =,, k =,,,, l =,,, 9a =,, b =,, c = , ,, a =,, b =,, c =, Eercise. a,, b,, c,, d, 9, e, a, b, a,, b,, c,, d, 9,, cm cm cm a =, b cm 9a s, s, b s, c No, the greatest height is m when t =. s. m = cm, 9 cm a, b, c = --, b cm cm cm cm a A =, b m m, m m -- Chapter Review a es, b es, c no, d es, e es, f no a =, b = + a = +, b = + am =,, b =,, c t =,, d n =, --, e p =,, f k = --, -- ap =,, b u =,, c n =,, d e =,, e =,, f c =,, g =, h a =,, i q = --,, j t = --, aa =,, b t =,, c b =,, d =,, e k =,, f w = --, a =,, b = ± a =.,., b =.9,.9, c =.,., d =.,. 9a =,, b = --,, ± 9 ± c = , d = a =,, b =.,., c =,, d =.,., e =,, f =,, g = --, a,, b,, c, 9 a cm cm, b cm = a s, b s, s, c s Graphs in the number plane Eercise. ar, b P, c P, d Q ab, b C, c A, E, d D, E ai Ton, ii Pete, iii Quentin, iv Robin, bi Quentin, Robin, ii Pete, Steve a A, D, b B, C c D, d B B C B D 9 A A A a B, b C, c A a D, b B, c C, d A

21 Answers a b c d Height Height Time Height Height Time Height Time Height Time a She probabl opened a new account, since the opening balance is $. b Tuesda, Wednesda, Frida, c Frida, d Thursda, e Sunda Time Time 9 a b Water level Water level Time (minutes) Time (minutes) Water level Noise level Jan. Feb. Mar. April Ma June Jul Time a b c d Distance Distance Distance Distance Time (s) Time (s) Time (s) Time (s) Eercise. a km, b Her speed increased, the line became steeper. c noon, d -- h, e km, f -- h, g km a 9 am, b km, ci km/h, ii km/h, d noon, e km, f km a km, b km, c am noon, d km, e km, f. pm, g km a km, b. pm pm, c. am, d. am, 9. am,. pm, e 9am 9. am,. am. am, f -- h, g km a am, b. am, 9. am, c km, d Steve, e km, f km, g Steve km/h, Perr km/h a km, b am,. pm, c km, d Sourav, e Irena, b km, f. pm. pm, g. km/h

22 Mathscape Etension a Distance (km) noon Time b km/h, c pm, d. pm, e. am,. pm a a Distance (km) 9 noon Time b h, c. pm, d km/h, e km Eercise. a b c = = + = d c + = = a d = = b e = + 9 =

23 Answers f g h = i j k = = = + + = = l = + am =, b =, b m =, b =, c m =, b =, d m =, b =, e m =, b =, f m =, b =, g m = --, b =, h m = --, b = a = +, b = +, c =, d = --, e = -- a = + --, m =, b = --, b = -- --, m = --, b = --, c = --, m = --, b =, d = , m = --, b = a b c = = + = +

24 Mathscape Etension d e f = = = g h i = = + = + j k l = = = a =, b = +, c =, d = a =, b = --, c =, d = -- 9a = +, b = +, c = +, d = --, e = --, f = -- a =, b =, c =, d =, e =, f =, g =, h = a =, b =, c =, d = a es, b no, c no, d es, e no, f es a no, b es, c es, d no, e es, f no, g es, h es a no, b es, c es, d no, e no, f es, g es, h no a c =, b g =, c m =, d z = a = +, b =, c = +, d =, e = +, f = 9, g = +, h = -- ai = 9, ii =, iii =, bi = 9, ii =,, iii =, c = + for, = + for < ai =, ii =, iii =, bit =, ii = --, --,, iii t, t =, 9