Coordinate Geometry
Gradients, Distances and Midpoints
● Gradients of lines can be found by dividing the change in y by the change
in x
● Parallel lines have equal gradients, perpendicular lines have negative
reciprocal gradients (e.g 2 and -½)
● The midpoint of a line segment can be found by using the formula:
x1 + x2 y 1 + y 2
M idpoint = ( 2 , 2 )
● The length of a line segment can be found by using pythagoras’ theorem:
Length =
√(x 1 − x2 )2 + (y 1 − y 2 )2
The Equation of a Straight Line
● Often in the form y = mx + c
● m is the gradient, c is the y-intercept
● If asked to give the equation of a line from given information, calculate the
gradient and y-intercept and the equation will be in the above form using
these two terms
● Occasionally you will only get the gradient and the coordinates of a single
point or two coordinates
● In this instance, use the equation y − y 1 = m(x − x1 )
● This is useful as you only need to know the gradient - you do not need to
calculate the y-intercept
The Intersection of Two Lines
● Point of intersection of two lines is found by solving the equations of the two
lines simultaneously
● This can give either the x or the y coordinate of the P.O.I
● To find the remaining coordinate, plug the coordinate you found into the
equation of either line
The Equation of a Circle
2 2
● The general form for the equation of a circle is (x − a) + (y − b) = r2
● The radius of the circle is r
● The coordinates of the centre of the circle are (a, b)
Finding the Equation of a Circle
● You can find the radius of a circle by calculating the distance between the
centre of the circle and a point on the circumference
● To find the centrepoint from three points on the circumference:
○ Find the bisectors of two of the lines joining two of the points
○ The centrepoint is the point of intersection of the two bisectors
Gradients, Distances and Midpoints
● Gradients of lines can be found by dividing the change in y by the change
in x
● Parallel lines have equal gradients, perpendicular lines have negative
reciprocal gradients (e.g 2 and -½)
● The midpoint of a line segment can be found by using the formula:
x1 + x2 y 1 + y 2
M idpoint = ( 2 , 2 )
● The length of a line segment can be found by using pythagoras’ theorem:
Length =
√(x 1 − x2 )2 + (y 1 − y 2 )2
The Equation of a Straight Line
● Often in the form y = mx + c
● m is the gradient, c is the y-intercept
● If asked to give the equation of a line from given information, calculate the
gradient and y-intercept and the equation will be in the above form using
these two terms
● Occasionally you will only get the gradient and the coordinates of a single
point or two coordinates
● In this instance, use the equation y − y 1 = m(x − x1 )
● This is useful as you only need to know the gradient - you do not need to
calculate the y-intercept
The Intersection of Two Lines
● Point of intersection of two lines is found by solving the equations of the two
lines simultaneously
● This can give either the x or the y coordinate of the P.O.I
● To find the remaining coordinate, plug the coordinate you found into the
equation of either line
The Equation of a Circle
2 2
● The general form for the equation of a circle is (x − a) + (y − b) = r2
● The radius of the circle is r
● The coordinates of the centre of the circle are (a, b)
Finding the Equation of a Circle
● You can find the radius of a circle by calculating the distance between the
centre of the circle and a point on the circumference
● To find the centrepoint from three points on the circumference:
○ Find the bisectors of two of the lines joining two of the points
○ The centrepoint is the point of intersection of the two bisectors
