COORDINATE GEOMETRY

MATHS

Graphs

Gradient of a Straight Line

• Measure of the steepness of a straight line

• Gradient=y2−y1x2−x1Gradient=x2​−x1​y2​−y1​​

Equation of a LineAx+By=C(General Equation of Line)Ax+By=C(General Equation of Line)xa+yb=1(Intercept Form)ax​+by​=1(Intercept Form)y=mx+c(Slope–Intercept Form)y=mx+c(Slope–Intercept Form)

• m = gradient

• c = y-intercept (the point the graph meets the y-axis)

  

Midpoint of GraphMidpoint=(x1+x22, y1+y22)Midpoint=(2x1​+x2​​, 2y1​+y2​​)Length between Two PointsDistance=(x2−x1)2+(y2−y1)2Distance=(x2​−x1​)2+(y2​−y1​)2​Cartesian Coordinate System

• They are written in the (x,y) for

  

• Quadrant 1: (x,y)

• Quadrant 2: (-x,y)

• Quadrant 3: (-x,-y)

• Quadrant 4: (x,-y)

Sketching Graphs

Graphs

• Step 1: Plot the y-intercept (c) on the graph

• Step 2: Substitute x values (whole number values) into the equation and solve for y

• Step 3: The substituted x value is the x coordinate, and the solved y value is the y coordinate

• Step 4: Repeat for 4 more values and draw a straight line connecting these points

How to Derive Equation of Line from Graph

Find the gradient of (0,7)(0,7) and (4,4)(4,4):

m=y2−y1x2−x1m=x2​−x1​y2​−y1​​

m=4−74−0m=4−04−7​

m=−34m=4−3​

Equation of the line: y=mx+cy=mx+c

y=−34x+cy=−43​x+c

Substitute (0,7)(0,7):

7=−34(0)+c7=−43​(0)+c

7=c7=c

Therefore, y=−34x+7y=−43​x+7

Parallel Lines and Perpendicular Bisectors

Parallel Lines

• Two lines are parallel if they share the same gradient but different y-intercepts

• For eg, y=3x+5 and y=3x+12y=3x+5 and y=3x+12

• Since the gradient of both lines is the same, and they have different y-intercepts, they are parallel to each other

Perpendicular Lines

• Two Lines are perpendicular if they meet at 90º

• If Line A has a gradient m, then the gradient of Line B is -1/m

• For eg, y=2x+4 and y=−12x+5y=2x+4 and y=2−1​x+5

• Two lines are perpendicular if the gradient of one line is the negative reciprocal of the other

Perpendicular Bisectors

• Perpendicular Bisector is a line that intersects another line at its midpoint, and lies at an angle of 90º

  

• Example:

  Point A has coordinates (7, −17)

  Point B has coordinates (19, −11)

  We want the equation of the perpendicular bisector of AB.

  Step 1: Midpoint

  M= (x1+x22, y1+y22) M=(2x1​+x2​​,2y1​+y2​​)

  M= (7+192, −17+(−11)2) M= (27+19​,2−17+(−11)​)

  M= (282) M= (226​,2−28​)

  M= (14) M13, −14)

  Step 2: Gradient of AB

  m AB=y2−y1x2−x1mAB​=x2​−x1​y2​−y1​​

  m AB=−11−(−17)19−7mAB​=19−7−11−(−17)​

  m AB=612=12mAB​=126​=21​

  Step 3: Gradient of perpendicular bisector

  m⊥=−1m AB=−2m⊥​=−m AB​1​=−2

  Step 4: Equation of perpendicular bisector

  y−(−14) =−2(x−13) y−(−14) =−2(x−13)

  y+14=−2x+26y+14=−2x+26

  y=−2x+12y=−2x+12

  Therefore, the equation of the perpendicular bisector is:

  y=−2x+12y=−2x+12​

COORDINATE

GEOMETRY