1 coordinate geometry - Maths Excel Class · 3. Line 1 passes through the points (6 ,9) and (5 ,3)....

Co-ordinateGeometry–Co-ordinates

©MathsExcelClass2015 3rdYrMary1

Everypointhastwoco-ordinates.

(3, 2)

Plotthefollowingpointsontheplane.

A(4, 1) D(2, −5) G(6, 3)B(3, 3) E(−4, −4) H(6, −3)C(−6, 5) F(−1, 2) I(−3, 1.5)

𝑥co-ordinate 𝑦co-ordinate

. (3, 2)

Co-ordinate Geometry – Gradient


Thegradient(𝑚)istheslopeoftheline.Positivegradient(𝒎 > 𝟎) Negativegradient(𝒎 < 𝟎)

𝒎 = 𝟎

Abiggergradientmakesasteeperline.

𝑚 = 1

𝑚 = 2

𝑚 = −1𝑚 = −2



Example1Findthegradientofthefollowingline.

𝑚 = 56

Example2Findthegradientoftheline.

𝑚 = 7897:;89;:

= <9=

>95

= ?

=

𝑚 = @ABC@DE

= 7897:

;89;:

.

(6, 7).

(3, 2)

Rise = 7 − 2 = 5

Run = 6 − 3 = 3

. (4, 3)

Rise=3

Run=4



QuestionsPlotthefollowingpointsandfindthegradient.

1. (5, 3)and(2, 1)

2. (7, 5)and(1, 1)

3. (3, 8)and(0, 0)

4. (5, 4)and(−2, 1)

5. (−3, −4)and(9, 1)

6. (−4, −1)and(−8, −2)

Findthegradientofthelineforeachpairofpoints.

1. (5, 8)and(1, 4)

2. (−3, 5)and(10, −2)

3. (10, 35)and(2, 3)

4. (−4, 7)and(−5, 1)

5. (22, 7)and(2, 2)

Co-ordinateGeometry–Equationofaline


Therearetwoformsfortheequationofaline.Generalform: 𝑎𝑥 + 𝑏𝑦 + 𝑐 = 0Gradient–Interceptform: 𝑦 = 𝑚𝑥 + 𝑏Therearethreewaystofindtheequationofalinedependingonwhatinformationyouhave.

1. Gradientandpoint

2. Twopoints

3. Gradientand𝑦-intercept TestingifapointisonthelineSubthepointintotheequation.Ifitsatisfiestheequation,thepointliesontheline.ExampleDoesthepoint(3,4)lieontheline2𝑥 − 3𝑦 + 6 = 0?LHS = 2 3 − 3 4 + 6 = 6 − 12 + 6 = 0 =RHS∴Thepoint(3,4)isontheline.

gradient 𝑦-intercept

𝒚 − 𝒚𝟏 = 𝒎(𝒙 − 𝒙𝟏)

𝒚 − 𝒚𝟏𝒙 − 𝒙𝟏

=𝒚𝟐 − 𝒚𝟏𝒙𝟐 − 𝒙𝟏

𝒚 = 𝒎𝒙 + 𝒃

𝑥

𝑦



𝒙-interceptsand𝒚-intercepts The𝒚-interceptis3.Theco-ordinatesare(0, 3).The𝒙-interceptis−4.Theco-ordinatesare(−4, 0).Example1Theequationofalineis3𝑥 − 4𝑦 + 12 = 0.Findthe𝑥-interceptand𝑦-intercept.𝒚-interceptSub𝑥 = 0intotheequation.3𝑥 − 4𝑦 + 12 = 03(𝟎) − 4𝑦 + 12 = 04𝑦 = 12𝑦 = 3 ∴The𝑦-interceptis3.𝒙-interceptSub𝑦 = 0intotheequation.3𝑥 − 4𝑦 + 12 = 03𝒙 − 4(𝟎) + 12 = 03𝑥 = −12𝑥 = −4 ∴The𝑥-interceptis−4.

𝑥

𝑦

3 −

−4

−



Findthe𝑥-interceptand𝑦-interceptofthefollowinglines.Sketchthegraphs.

1. 5𝑥 − 2𝑦 + 10 = 0

2. 3𝑥 − 2𝑦 + 6 = 0

3. 10𝑥 − 𝑦 + 10 = 0

4. 2𝑦 − 7𝑥 + 14 = 0

5. 𝑦 − 2𝑥 + 6 = 0

6. 𝑦 = T=𝑥 + 4

7. 𝑦 = =5𝑥 − 12

8. 𝑦 = 3𝑥 − 11



Findtheequationofthefollowinglinesingradient-interceptform.

1. Thelinehasagradientof2andpassesthroughthepoint(3, 4).

2. Thelinehasagradientof7andpassesthroughthepoint(−5, 2).

3. ThelinehasagradientofT=andpassesthroughthepoint(1, 4).

4. Thelinepassesthroughthepoints(1, 3)and(6, 8).

5. Thelinepassesthroughthepoints(6, 2)and(9, 3).

6. Thelinehasagradientof5?anda𝑦-interceptof4.



Findtheequationofthefollowinglinesingeneralform.

1. Thelinehasagradientof?>andpassesthroughthepoint(1, 2).

2. Thelinehasagradientof=Uandpassesthroughthepoint(5, −3).

3. Thelinepassesthroughthepoints(3, −1)and(5, 6).

4. Thelinepassesthroughthepoints(2, 11)and(−3, −1).

5. ThelinehasagradientofT6andan𝑥-interceptof8.



1. Showthatthepoint(0, 2)liesontheline𝑥 − 4𝑦 + 8 = 0.

2. Showthatthepoint(−2, 4)liesontheline𝑦 = 8𝑥 + 20.

3. Showthatthepoint(1, −7)liesontheline𝑦 = 4𝑥 − 11.

Co-ordinate Geometry – Sketching


Fortheequation𝒚 = 𝟐𝒙 + 𝟑fillinthetableofvalues.𝑦

𝑥 −4 −3 −2 −1 0 1 2 3 4

Sketchthegraph.Remember:

• Alwayslabelthe𝑥and𝑦axeswithbothletterandarrow

• Alwaysputanarrowateachendofthelinesketched

• Labelthe𝑥-interceptand𝑦-interceptifpossibleTosketchagraph,youonlyneedtoknowtwopoints.Atableofvaluesisnotnecessary.Changingtheequationintogradient-interceptformisuseful.



Example1Sketchthelinethatpassesthroughthepoints(1, 4)and(5, 6).Plotthepoints. Example2Sketchthegraphof2𝑥 − 3𝑦 + 6 = 0.𝑥-intercept:−3𝑦-intercept:2Example3Sketchthegraphof𝑦 = 3

4 𝑥 + 2.𝑦-intercept:2Gradient=@ABC

@DE= 5

6

. . (1, 4)

(5, 6) . . (1, 4)

(5, 6)

−

−

2

−3

−

2𝑟𝑢𝑛 = 4

𝑟𝑖𝑠𝑒 = 3

. (4, 5)

−

2

. (4, 5)



1. Sketchthelinethatpassesthroughthepoints(3, 5)and(9, 7).Whatisthegradient?

2. Sketchthelinethatpassesthroughthepoints(−6, −5)and(2, 5).Whatisthegradient?

3. Sketchthelinethatpassesthroughthepoints(1, −8)and(10, 1).Whatisthegradient?



4. Graphthelinewithgradient=5anda𝑦-interceptof4.

5. Graphthelinewithgradient−52anda𝑦-interceptof1.

6. Graphthelinewithgradient]Uanda𝑦-interceptof−3.



7. Sketchthelinewithequation𝑦 = 14 𝑥 + 5.

8. Graphthelinearfunction𝑦 = −2𝑥 + 6.

9. Graphthelinearfunction𝑦 = −3𝑥 − 4.



10. Sketchthelinewithequation3𝑥 − 2𝑦 − 12 = 0.

11. Graphthelinearfunction8𝑥 + 3𝑦 − 24 = 0.

12. Graphthelinearfunction4𝑥 + 3𝑦 + 12 = 0.

Co-ordinate Geometry – Parallel lines


Example1IsLine1paralleltoLine2?Line1passesthrough Line2passesthrough(2, 6)and(0, 1). (4, 5)and(2, 0).

𝑚T =6−12−0 =

?= 𝑚= =

5−04−2 =

?=

𝑚T = 𝑚= ∴Line1isparalleltoLine2.Example2Arethelinesparallel?Line1:𝑦 = 1

4 𝑥 + 6. Line2:𝑥 − 4𝑦 + 23 = 0 𝒎𝟏 =

𝟏𝟒 4𝑦 = 𝑥 + 23

𝑦 = 1

4 𝑥 +234

𝒎𝟐 =

𝟏𝟒

𝒎𝟏 = 𝒎𝟐∴Thelinesareparallel.

Parallellines:Gradientsarethesame

𝒎𝟏 = 𝒎𝟐

−1−

2

(4, 5). . (2, 6)Line1

Line2

Co-ordinate Geometry – Parallel lines


1. Line1passesthroughthepoints(4, 5)and(2, 2).Line2passesthroughthepoints(8, 7)and(6, 4).Sketchbothlinesonthesameplaneandprovetheyareparallel.

2. LineAhastheequation𝑥 − 7𝑦 − 33 = 0.LineBisparalleltoLineAandpassesthroughthepoint(5, 1).FindtheequationofLineBandsketchbothlinesonthesameplane.

3. Provethefollowinglinesareparallelandsketcheachpaironthesameplane.

a. Line1:𝑦 = 45 𝑥 + 1

Line2:4𝑥 − 5𝑦 + 20 = 0

b. Line1:7𝑥 + 3𝑦 + 21 = 0

Line2:7𝑥 + 3𝑦 − 42 = 0

Co-ordinate Geometry – Perpendicular lines


Perpendicularlinesintersectatanangleof90°.𝑨𝑩 ⊥ 𝑪𝑫.Example1IsLine1perpendiculartoLine2?Line1hasequation Line2hasequation4𝑥 − 3𝑦 + 3 = 0. 3𝑥 + 4𝑦 − 16 = 0.ChangetoG-Iform. 𝑦 = 4

3 𝑥 + 1 𝑦 = −34 𝑥 + 4

𝑚T= 43 𝑚== −3

4𝒎𝟏×𝒎𝟐 =

𝟒𝟑×− 𝟑

𝟒= −𝟏 ∴Line1isperpendiculartoLine2.

−1

−4

Line1:4𝑥 − 3𝑦 + 3 = 0

Line2:3𝑥 + 4𝑦 − 16 = 0

Perpendicularlines:Gradientsarethenegativereciprocalofeachother

𝒎𝟏 =−𝟏𝒎𝟐or𝒎𝟏𝒎𝟐 = −𝟏

A

BC

D

Co-ordinate Geometry – Perpendicular lines


1. Writethenegativereciprocalofthefollowing:

a. =5

b. −>

<

c. 3

d. −5

2. Line1passesthroughthepoints(3, 7)and(1, 4).Line2passesthroughthepoints(3, 3)and(0, 5).ProvethatLine1isperpendiculartoLine2andsketchbothlinesonthesameplane.

3. Line1passesthroughthepoints(6, 9)and(5, 3).Line2passesthroughthepoints(8, 4)and(2, 5).ProvethatLine1isperpendiculartoLine2andsketchbothlinesonthesameplane.

4. LineAhastheequation2𝑥 − 5𝑦 + 1 = 0.LineBisperpendiculartoLineAandpassesthroughthepoint(3, 4).FindtheequationofLineBandsketchbothlinesonthesameplane.

Co-ordinate Geometry – Midpoint


Midpoint:findtheaverageofeachco-ordinate.

𝑥co-ordinate=;:f;8= 𝑦co-ordinate=

7:f78=

Example1Findthemidpointbetween(1, 2)and(5, 8).Midpoint:

𝑥co-ordinate=Tf?= 𝑦co-ordinate=

=f]=

= 3 = 5∴Themidpointis(3, 5).

. (5, 8)

. (1, 2)

. (5, 8)

. (1, 2)

. (3, 5)

g𝑥T + 𝑥=2

,𝑦T + 𝑦=2

h

Co-ordinate Geometry – Midpoint


Plotthefollowingpointsandfindthemidpoint.

1. (3, 4)and(7, 8)

2. 2, 6 and 4, 10

3. (5, 1)and(1, −9)

4. PointAisthe𝑦-interceptoftheline𝑦 = 2𝑥 + 3.PointBhasco-ordinates4, 11 .

a. ProvethatPointBliesontheline𝑦 = 2𝑥 + 3.

b. Sketchtheline𝑦 = 2𝑥 + 3indicatingPointAandPointB.FindthemidpointofAandBandaddittothediagram.

Co-ordinate Geometry – Distance


ExampleFindthedistance(D)betweenthepoints(2, 4)and(6, 7).Disthehypotenuseofaright-angledtriangle.Theheightofthetriangleis(7 − 4).Thebaseofthetriangleis(6 − 2).UsingPythagoras’theorem:𝐷= = 6 − 2 = + 7 − 4 = 𝐷 = 6 − 2 = + 7 − 4 = 𝐷 = 4= + 3= = 16 + 9 = 25 𝐷 = 5𝑢𝑛𝑖𝑡𝑠

Distance= k(𝒙𝟐 − 𝒙𝟏)𝟐 + (𝒚𝟐 − 𝒚𝟏)𝟐units

. (6, 7).

(2, 4)D

. (6, 7).

(2, 4)Height= 7 − 4

Base= 6 − 2

D



1. PointAhasco-ordinates 2, 7 andPointBhasco-ordinates 8, 15 .PlotthepointsandfindthedistanceoflineAB.

2. PointAhasco-ordinates 4, 1 andPointBhasco-ordinates 8, 7 .PlotthepointsandfindthedistanceoflineAB.

3. PointAhasco-ordinates 11, 10 andPointBhasco-ordinates 6, −2 .PlotthepointsandfindthedistanceoflineAB.

4. PointAhasco-ordinates −9, −7 andPointBhasco-ordinates −2, −6 .PlotthepointsandfindthedistanceoflineAB.



5. PointAhasco-ordinates 4, 0 andPointBhasco-ordinates 6, −10 .PlotthepointsandfindthedistanceoflineAB.

6. PointAhasco-ordinates 2, 2 andPointBhasco-ordinates −3, 1 .PlotthepointsandfindthedistanceoflineAB.

7. Onamap,aredflagisplacedatco-ordinates −5, 8 andagreenflagisplacedatco-ordinates −8, −1 .Plotthepositionofthetwoflagsandfindthedifferencebetweenthem.

8. PointAisthemidpointbetween −3, 8 and 5, 4 .PointBhasco-ordinates 13, 12 .WhatisthedistanceoflineAB?



9. PlotpointA −3, 2 ,pointB 5, 2 andpointC 5, 8 .

a. FindthedistanceofAB.

b. FindthedistanceofBC.

c. UsingeitherthedistanceformulaorPythagoras’theorem,findthedistanceofAC.

d. Findtheareaof△ 𝐴𝐵𝐶.

e. FindtheequationoflineAC.

f. FindtheequationsoflinesABandBC.

Co-ordinate Geometry – Perpendicular Distance


Perpendiculardistanceistheshortestdistancebetweenapointandaline.Example1Findtheperpendiculardistancebetweenthepoint 4, 1 andtheline2𝑥 − 3𝑦 + 6 = 0.

Distance =𝑎𝑥1+𝑏𝑦1+𝑐

𝑎2+𝑏2

Fromtheequation2𝑥 − 3𝑦 + 6 = 0. 𝑎 = 2 𝑏 = −3 𝑐 = 6Fromthepoint 4, 1 𝑥T = 4 𝑦T = 1

Distance =2 4 −3 1 +6

22+(−3)2 =

8–3+64+9

=1113 = 11

13units

= 11 1313 units(rationaldenominator)

(4, 1).

2𝑥 − 3𝑦 + 6 = 0

D

Perpendiculardistance= |𝒂𝒙𝟏f𝒃𝒚𝟏f𝒄|k𝒂𝟐f𝒃𝟐

units



Drawintheperpendiculardistancebetweenthefollowinglinesandpoints.

. .

.

. REMEMBER:

• Tousetheperpendiculardistanceformula,theequationofthelinemustbeingeneralform.

• Arrangetheequationsothatthe𝑥comesbeforethe𝑦.



1. Findtheperpendiculardistancebetweenthepoint 1, 8 andtheline4𝑥 + 3𝑦 + 1 = 0.

2. Findtheperpendiculardistancebetweenthepoint 3, −2 andtheline8𝑥 + 6𝑦 − 5 = 0.

3. Findtheperpendiculardistancebetweenthepoint −5, 1 andtheline𝑥 − 7𝑦 − 10 = 0.

4. Findtheperpendiculardistancebetweenthepoint 3, 5 andtheline2𝑦 + 7𝑥 + 4 = 0.



5. Findtheperpendiculardistancebetweenthepoint 4, 2 andtheline𝑦 =35 𝑥 + 3.

6. Sketchtheline𝑦 = −14 𝑥 − 2andfindtheperpendiculardistancefromthe

point 4, 2 totheline.

7. Sketchtheline𝑦 = 43 𝑥 − 11andfindtheperpendiculardistancefromthe

point 1, 2 totheline.

1 coordinate geometry - Maths Excel Class · 3. Line 1 passes through the points (6 ,9) and (5 ,3)....

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Transcript of 1 coordinate geometry - Maths Excel Class · 3. Line 1 passes through the points (6 ,9) and (5 ,3)....