Finding Distance in the Finding Distance in the Cartesian PlaneCartesian Plane

The Cartesian PlaneThe Cartesian Planeis the x-y Graph!is the x-y Graph!

Distance Between 2 PointsDistance Between 2 PointsDistance between points can be Distance between points can be determined for the 3 cases:determined for the 3 cases:Horizontal distanceHorizontal distanceVertical distanceVertical distanceOblique (diagonal distance)Oblique (diagonal distance)N.B. Distance is always positive.N.B. Distance is always positive.

Distance TypesDistance TypesThe triangle ABC has:The triangle ABC has:

Side BC: horizontalSide BC: horizontalSide AB: verticalSide AB: verticalSide AB: obliqueSide AB: oblique

Horizontal DistanceHorizontal DistanceIn this situation, the two points have In this situation, the two points have the same y coordinate, so to find the the same y coordinate, so to find the distance, subtract the x coordinates.distance, subtract the x coordinates.

E.g (8,0) – (4,0) = 4E.g (8,0) – (4,0) = 4E.g.(8,1) – (-4,1) = 8 - -4 = 12E.g.(8,1) – (-4,1) = 8 - -4 = 12

Vertical DistanceVertical DistanceIn this situation, the two points have In this situation, the two points have the same x coordinate, so to find the the same x coordinate, so to find the distance, subtract the y coordinates.distance, subtract the y coordinates.

E.g (0,7) – (0,3) = 4E.g (0,7) – (0,3) = 4E.g.(8,1) – (8,-2) = 1--2 = 3E.g.(8,1) – (8,-2) = 1--2 = 3

Vertical/Horizontal DistanceVertical/Horizontal Distance

Oblique DistanceOblique DistanceIn this situation, the two points have In this situation, the two points have the different x and y coordinates.the different x and y coordinates.We need Pythagoras!We need Pythagoras!cc22 = a = a22 + b + b22

c – hypotenuse, thec – hypotenuse, theside opposite the 90side opposite the 90˚ ˚ and the longest side.and the longest side.

a, b the other legsa, b the other legs

Pythagoras to the Rescue!Pythagoras to the Rescue!AB hypotenuseAB hypotenuseAC, BC legsAC, BC legsAC = 7 squaresAC = 7 squaresBC = 3 squaresBC = 3 squaresAB = how long?AB = how long?

Distance FormulaDistance FormulaThe distance between two points can The distance between two points can be found using the distance formula.be found using the distance formula.The distance between (The distance between (xx11, , yy11) and () and (xx22, , yy22) is given by: ) is given by: Distance, d = Distance, d = √(x√(x22-x-x11))22 + (y + (y22-y-y11))22

Use: d = Use: d = √(x√(x22-x-x11))22 + (y + (y22-y-y11))22

Determine: Determine: d (0,0) to Ad (0,0) to Ad (0,0) to Bd (0,0) to Bd (A,B)d (A,B)

Midpoint of a LineMidpoint of a LineGiven 2 points (xGiven 2 points (x11, y, y11) and (x) and (x22,y,y22) the ) the middle point or “midpoint” can be middle point or “midpoint” can be determined by the following:determined by the following:Midpoint (x,y) = (Midpoint (x,y) = (xx11+x+x22, , yy11+y+y22) ) 2 2 2 2Find the midpoint of (5,7) & (11,29)Find the midpoint of (5,7) & (11,29)Find the midpoint of (-3,-5) & (17, 12)Find the midpoint of (-3,-5) & (17, 12)

Exam QuestionExam QuestionTo service a new residential development, the town surveyor has drawn on a Cartesian plane the new part of the water main that must be constructed. DE represents the existing water main. FG and GM represent the new water main, where M is the midpoint of DE

1

1

M

D(0, 4)

E(3, -2)

F(-3, -4)

G(-5, -1)

y

x

Rounded to the nearest tenth, what is the total length of the new water main FGM? Show all your work.

ActivitiesActivitiesP.152P.152Questions 1abcd, 2, 3 ,7Questions 1abcd, 2, 3 ,7