Coordinte Geometr Coordinte Geometr Curricuum Red ACMNA: 78, 4, 94 www.mthetics.com
Coordinte COORDINATE Geometr GEOMETRY Shpes ou ve seen in geometr re put onto es nd nsed using gebr. Epect bit of both in this section. Answer these questions, before working through the chpter. I used to think: Wht do we men when we spek of "coordintes of point"? How do ou find the midpoint between n two points using on their coordintes? Wht is the sope of ine? Answer these questions, fter working through the chpter. But now I think: Wht do we men when we spek of "coordintes of point"? How do ou find the midpoint between n two points using on their coordintes? Wht is the sope of ine? Wht do I know now tht I didn t know before? 00% Coordinte Geometr Mthetics 00%
Coordinte Geometr Bsics Sstem of Aes A sstem of es is horizont number ine ced the -is nd vertic number ine ced the -is. for -is 6 5 4 -is (vertic) B on -is, on -is -6-5 -4 - - - 0 - - -is (horizont) 4 5 6 Lines ws cross t zero for -is - -4-5 -6 The point where the number ines cross is ced the origin. Points hve positions on both number ines. Above, the point beed B, hs position of on the -is (horizont number ine) nd on the -is (vertic number ine). Coordintes (,) Ech point hs co-ordintes bsed on where it ies on the es. 4 A B O -4 - - - 0 4 - D - E - -4 A is in ine with - on the -is nd on the -is. So, A hs coordintes (-, ). The -coordinte is ws written first. B hs coordintes (, ) D hs coordintes (, -) E hs coordintes (-, -) O is the origin nd hs coordintes (0, 0) 00% Coordinte Geometr Mthetics 00%
Coordinte Geometr Bsics Distnce Between Points on the Sme Horizont or Vertic Line The distnce between points on the sme horizont or vertic ine is found b simp counting the units between them. B( -,) 4 A(,) C(,) D(4,) -4 - - - 0 4 - - F(, -) - -4 Distnce AC = units Distnce BD = 7 Units Distnce AF = Distnce Between An Two Points To find the distnce between two points which re not on the sme ine, we use Pthgors s Theorem. A( -,4) 4 6 units C(,) -4 - - - 0-4 - - -4 B( 4, -4) 8 units Find the ength of the ine AB Step : Step : Step : Compete the right nged tringe (gre ine) Find the vertic nd horizont distnces of the right nged tringe Use Pthgors s theorem with ine AB s the hpotenuse AB = 6 + 8 = 00 AB = 00 = 0 units Wht re the coordintes of C? Compete to find AC: AC = + = AC = AC = 00% Coordinte Geometr Mthetics 00%
Coordinte Geometr Questions Bsics Use the sstem of es beow to nswer the questions tht foow: 6 5 4-6 -5-4 - - - 0 4 5 6 - - - -4-5 -6. Use highighter to identif the -is, -is nd origin.. Pot the foowing points A G on the bove set of es. A^, h B^, h C^04, h D^-, h E^-5, -4h F TheOrigin G^0, h H^5, -5h. Find the engths of the foowing ines: AB b BG c FC d From theoriginto (5,0) 4 00% Coordinte Geometr Mthetics 00%
Coordinte Geometr Questions Bsics 4. Find the engths of the foowing ines: CG b DH 5. Find the engths of the foowing ines (eve our nswers in surd form): Theoriginto E b DE 00% Coordinte Geometr Mthetics 00% 5
Coordinte Geometr Knowing More Distnce Formu Let s use wht we know bout Pthgors s Theorem to find formu which ccutes the distnce between n two points. Let point A ^, h nd B ^, h be n points. Wtch how Pthgors s theorem is used to find the distnce d. B (, ) - - A (, ) - (, ) - Compete the tringe. The horizont distnce is - nd the vertic distnce is -. Use Pthgors, ike before AB = ( - ) + ( -) AB = ( - ) + ( -) This is ced the distnce formu. It cn be used to ccute the distnce between n two points ^, h nd ^, h. Find the distnce, d, from ^-, 7h to ^9, h d = ( - ) + ( - ) d = ( 9-( - )) + ( - 7) d = ( ) + (- 5) = 69 = units Wht hppens if we switch the points round? Find the distnce, d, from ^9, h to ^-, 7h d = ( - ) + ( - ) d = (-- 9) + ( 7- ) d = (- ) + ( 5) = 69 = units It doesn t mtter which point ou choose s first nd second, the formu wi ws work. 6 00% Coordinte Geometr Mthetics 00%
Coordinte Geometr Knowing More The Midpoint of Line B (, ) - M (,) - A (, ) - - In the bove digrm, M is the midpoint on the ine AB. This is how to find the coordintes of M: -coordinte: must be in the midde of nd -coordinte: must be in the midde of nd ` - = - ` = + ` = + ` - = - ` = + ` = + So, the coordintes of the midpoint between n two points ^, h nd ^, h is simp the verge of the coordintes Find the midpoint between (,8) nd (-7, ) = + ( 7) = +- =- The midpoint hs coordintes (- 0, ). = + = 8 + = 0 Wht hppens if we switch the points round? Find the midpoint between (- 7, ) nd (,8) = + = - 7 + =- The midpoint hs coordintes (- 0, ). = + = 8 0 + = It doesn t mtter which point ou choose s first nd second, the formu wi ws work. 00% Coordinte Geometr Mthetics 00% 7
Coordinte Geometr Questions Knowing More. Find the distnce between the foowing points. Leve nswers s squre roots if necessr. (Hint: Distnce Formu) c ^-, 5h nd ^9, h b ^0, h nd ^-75, h c TheOrigin nd ^, 8h d ^-4, h nd ^-, 6h. Find the coordintes of the midpoint between: ^, h nd ^8, h b ^-, 5h nd ^7, h 8 00% Coordinte Geometr Mthetics 00%
Coordinte Geometr Questions Knowing More. Use the distnce formu to determine which of A^5, h or B^6, h is furthest from C^, h? (Hint: Pot the points on set of es first.) 4. The midpoint of the ine EF is ^5, -6h. Find the coordintes of E^, h if F^0, -0h. 00% Coordinte Geometr Mthetics 00% 9
Coordinte Geometr Questions Knowing More 5. A circe hs its centre t ^8, h. At one end of dimeter is the point ^-4, -7h. Find the coordintes of the point t the other side of the sme dimeter. (Hint: The centre is the midpoint of the dimeter) ( 8, ) -4 8 (-4,-7) -7 b Find the ength of the dimeter. 0 00% Coordinte Geometr Mthetics 00%
Coordinte Geometr Using Our Knowedge Grdient of Line The grdient of ine mesures how steep or gente the sope of the ine is. It cn be found in two ws: the grdient formu, or using the nge the ine mkes with the positive -is. Grdient Formu Before we ern the formu we must first define "run" nd "rise" (or "f"). "Run" is the horizont distnce ine trves from eft to right. "Rise" is the distnce the ine trves upwrds. "F" is the distnce ine trves downwrds. B (, ) A (, ) run rise f A (, ) run B (, ) As with the previous section, the run is - nd the rise is -. The formu for grdient, smboized b m is: Grdient rise - = m = = run - Find the grdient of the ine which psses through the points ^-, 6h nd ^40, h - m = = 0-6 - 4 -- ( ) = 4 = 6 Wht hppens if the points re switched round? Find the grdient of the ine which psses through the points ^40, h nd ^-, 6h - m = = 6-0 - - - 4 = - 4 = - 6 It doesn t mtter which point ou choose s first nd second, the formu wi ws work. 00% Coordinte Geometr Mthetics 00%
Coordinte Geometr Using Our Knowedge Grdient from the Ange If θ is the nge between the ine nd the positive -is, then m = tnθ. This is becuse in right nged tringe: rise θ run rise opposite m = = = tn i run djcent i is ced the nge of incintion. It is the nge between the ine nd the positive -is. Find the grdient of the two ines beow 45c 5c i = 45c i = 5c m = tni = tn 45c m = tni = tn 5c m = m =- Find the nge of incintion of ine which hs grdient of 6 to the nerest degree. 7 tn i = m tn i = 6 ` j 7 - i = tn 6 ` j = 0.65... c. c 7 (nerest degree) 00% Coordinte Geometr Mthetics 00%
Coordinte Geometr Using Our Knowedge Wht hppens if i turns out to be negtive? Find i if the grdient of ine is m =- tn i = m tn i =- i = tn - ^- h=-0c is this correct? 50c -0c Look t the ine bove. It mkes two nges with the positive -is. As rue, i is ws the positive nge. So, if the nge is negtive, simp dd 80c to the nswer. For the bove empe, the nge of incintion is - 0c+ 80c = 50c. Find the nge of incintion of ine which hs grdient of m =- to the nerest degree. tn i = m tn i =- i = tn - ^- h=-7565.... c Since i is negtive, dd 80c Ange of incintion =- 7.565... c+ 80c = 08. 44... c. 08c (nerest degree) This wi on be necessr when the grdient, m, is negtive. Wh? 00% Coordinte Geometr Mthetics 00%
Coordinte Geometr Using Our Knowedge Positive nd Negtive Sope The horizont run of ine is ws positive, since b definition it is horizont distnce trveed s the ine moves from eft to right. But, the rise m be positive or negtive (we c this "f"). Positive Rise Negtive Rise B (, ) A (, ) run rise f A (, ) run B (, ) When the rise is positive, the grdient of the ine is positive. The ine ens to the right since the ine rises. When the rise is negtive, the grdient of the ine is negtive. The ine ens to the eft since the ine fs. Positive grdient Negtive grdient Identif the ines in the foowing digrm s hving positive grdient or negtive grdient B C A E F D Positive grdient: AB, BC, DF Negtive grdient: AD, BE, EF 4 00% Coordinte Geometr Mthetics 00%
Coordinte Geometr Questions Using Our Knowedge. Find the grdient of the ines which pss through the foowing points: ^4, h nd ^64, h b ^66, h nd ^0, 8h c ^4, - 5h nd ^0, h d ^-, -7h nd ^-6, -4h. A ine psses through the points ^-, -h nd ^6, h. The grdient of the ine is m =-. Wht is the vue of?. A ine psses through the points ^, 9h nd ^, h. If the grdient of this ine is m =, then wht is the vue of? 00% Coordinte Geometr Mthetics 00% 5
Coordinte Geometr Questions Using Our Knowedge 4. Find the grdient of the foowing ines (to two decim pces): b 5c 80c c d 0c 50c Hint: Find the nge of incintion first! 6 00% Coordinte Geometr Mthetics 00%
Coordinte Geometr Questions Using Our Knowedge 5. Find the nge of incintion (to two decim pces) of ine which hs grdient: m = b m =-.5 c m = d m =- 6. Find the grdient of ine (to one decim pce) with nge of incintion: 40c b 70c 00% Coordinte Geometr Mthetics 00% 7
Coordinte Geometr Questions Using Our Knowedge 7. A ine is joined between A^, - h nd B^, h. Find the vue of if =- nd AB hs grdient m =-. 5 b Find the vue of if = nd AB hs grdient m =-. 4 c Find the vues of nd if the midpoint of ine AB is ^-6, 7h. 8 00% Coordinte Geometr Mthetics 00%
Coordinte Geometr Thinking More 5 4 B -6-5 -4 - - - 4 5 6 D - - - -4 C -5 Find the perimeter nd re of the bove squre ABCD: Use the distnce formu to find the ength of side AB: A^0, h nd B^40, h. AB = d = ( - ) + ( - ) = ( 0- ) + ( 0-4) = 9+ 6 = 5 = 5 units The perimeter of the squre is thus 4# 5 = 0 units. The re of the squre is 5# 5 = 5 units. Use the formu for midpoint to show tht the digons AC nd BD bisect ech other Find the Midpoint of A^0, h nd C^, -4h. -coordinte: -coordinte: + = 0+ = + +- ( 4) = =- 4 44 4 Midpoint of AC is, ` - j Find the Midpoint of B^40, h nd D^-, -h. -coordinte: -coordinte: + 4+- ( ) = = + 0+- ( ) = =- 4 44 4 Midpoint of BD is, ` - j AC nd BD hve the sme midpoints nd therefore the bisect ech other. 00% Coordinte Geometr Mthetics 00% 9
Coordinte Geometr Thinking More Three points re coiner (ie on the sme ine) if two of the ines connecting them hve the sme grdient. Ie, A, B nd C re coiner if either or occurs: 44444444 Sme grdient to BC 44444444 Sme grdient to AB B 444 4444 4 Different grdient to AB C D A A, B nd C re coiner, since the grdient of AB nd the grdient of BC re the sme. A, B nd D re not coiner. Are A^, h, B^-7, h nd C^6, - 9h on the sme ine (coiner)? - Grdient of AB: m = = 7- = 6 =- - - - - - Grdient of BC: m = = -9-7 = - 6 =- - 6 -- ( ) 8 Thus the grdient of AB is equ to the grdient of BC. Therefore ABC is stright ine nd points re coiner. The foowing points re coiner: D^-, -h, E^, h nd F^49, h. Sove for. Grdient of DE = Grdient of DF -- ( ) 9-- ( ) = -- ( ) 4 -- ( ) 6 + = 6 + = ` = 0 00% Coordinte Geometr Mthetics 00%
Coordinte Geometr Thinking More A ine cuts the -is with grdient is rotted 5c to the right. Wht is its new grdient? Find the nge of incintion before it is rotted. tn i = m = i = tn - ^ h= 60c Find the nge of incintion fter it is rotted. After rotting to 5c to the right, i = 60c- 5c = 45c 45c 60c Find the new grdient using the new nge of incintion. m = tni = tn 45c = A ine with nge of incintion Wht is the nge of rottion? i = 5c is rotted to the right to hve grdient m =. m - 5c Step : Find the nge of incintion fter rottion. m = Step : Find the difference between the origin nge nd the new nge. i = tn - ^h= 45c Ange of rottion = 5c- 45c = 90c 00% Coordinte Geometr Mthetics 00%
Coordinte Geometr Questions Thinking More. The preogrm ABCD hs vertices A^, h, B^0, 8h, C^8, h nd D^-, -5h. Do digons AC nd BD bisect ech other? b Find the perimeter of this preogrm. 00% Coordinte Geometr Mthetics 00%
Coordinte Geometr Questions Thinking More. Which set of points is not coiner? ^, h, ^-, -h, ^-, -7h ^640, h, ^, 6h, ^54, h ^4, -h, ^-4, h, ^0, h. The foowing points re coiner: L^-5, h, M^4, h nd N^, h. Sove for. 00% Coordinte Geometr Mthetics 00%
Coordinte Geometr Questions Thinking More 4. The foowing three points re vertices of tringe: P^-5, -4h, Q^-, h nd R^, -4h. Use the distnce formu to determine the tpe of tringe. 5. A certin ine hs grdient m =. Answer the foowing correct to two decim pces. Find i, the nge of incintion. b If the ine is rotted 0c to the eft, wi the grdient be positive or negtive? Find the new grdient. c If the origin ine is rotted 0c to the right, wi the grdient be positive or negtive? Find the new grdient. 4 00% Coordinte Geometr Mthetics 00%
00% Coordinte Geometr Mthetics 00% Coordinte Geometr 5 Questions Thinking More b 6. In regur cock fce: Identif between which times the grdient of the hour hnd wi be positive. Identif between which times the grdient of the hour hnd wi be negtive. 0 4 5 6 7 8 9
Coordinte Geometr Answers Bsics: Using Our Knowedge:. &.. b c - d 5 D (-,) C (0,4). 0 =- A (,) B (,) F (0,0) G (,0). 5 = 4. m. 0.70 ( d.p.) b m. 5.68 ( d.p.) E (-5,-4) H (5,-5) c m. -.7 ( d.p.) d m. -.9 ( d.p.) 5. 6. 57c b 69c.. AB = b BG = c 0c d 50c c FC 4 = d Length =5 6. m. 0.8 ( d.p.) b m..7 ( d.p.) 4. CG = 5 b DH = 0 7. =-4 b = 0 5. OE = 4 b DE = 65 c =-4 Knowing More: = 7. 5 b 7 Thinking More: c 85 d 5. Yes, the bisect ech other.. (5,0) b (,9) b Perimeter = 0 + 4 0 units. B is furthest from C. ^, h, ^-, -h, ^-, -7h re coiner. ^640, h, ^, 6h, ^54, h re not coiner. 4. E^0, -h ^4, -h, ^-4, h, ^0, h re not coiner. 5. (0,) b d = 4 6. 9 =- 6 00% Coordinte Geometr Mthetics 00%
Coordinte Geometr Answers Thinking More: 4. It is n isoscees tringe 5. b c θ = 0c m =.9 ( d.p.) So the grdient is positive. m = 0.8 ( d.p.) So the grdient is positive. 6. b When the time is between o'cock nd o'cock, but ecude o'cock nd o'cock. And when the time is between 6 o'cock nd 9 o'cock, but ecude 6 o'cock nd 9 o'cock. When the time is between o'cock nd 6 o'cock, but ecude o'cock nd 6 o'cock. And when the time is between 9 o'cock nd o'cock, but ecude 9 o'cock nd o'cock. 00% Coordinte Geometr Mthetics 00% 7
Coordinte Geometr Notes 8 00% Coordinte Geometr Mthetics 00%
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