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Coordinate Geometry (DP IB Maths: AI HL)
Revision Note
Basic Coordinate Geometry
What are cartesian coordinates?
- Cartesian coordinates are basically the x-y coordinate system
- They allow us to label where things are in a two-dimensional plane
- In the 2D cartesian system, the horizontal axis is labelled x and the vertical axis is labelled y
What can we do with coordinates?
- If we have two points with coordinates (x1 , y1) and (x2 , y2) then we should be able to find
- The midpoint of the two points
- The distance between the two points
- The gradient of the line between them
How do I find the midpoint of two points?
- The midpoint is the average (middle) point
- It can be found by finding the middle of the x-coordinates and the middle of the y-coordinates
- The coordinates of the midpoint will be
-
- This is given in the formula booklet under the prior learning section at the beginning
How do I find the distance between two points?
- The distance between two points with coordinates (x1 , y1) and (x2 , y2) can be found using the formula
-
- This is given in the formula booklet in the prior learning section at the beginning
- Pythagoras’ Theorem is used to find the length of a line between two coordinates
- If the coordinates are labelled A and B then the line segment between them is written with the notation [AB]
How do I find the gradient of the line between two points?
- The gradient of a line between two points with coordinates (x1 , y1) and (x2 , y2) can be found using the formula
-
- This is given in the formula booklet under section 2.1 Gradient formula
- This is usually known as
Worked example
Point A has coordinates (3, -4) and point B has coordinates (-5, 2).
i)
Calculate the distance of the line segment AB.
ii)
Find the gradient of the line connecting points A and B.
iii)
Find the midpoint of [AB ] .
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Perpendicular Bisectors
What is a perpendicular bisector?
- A perpendicular bisector of a line segment cuts the line segment in half at a right angle
- Perpendicular lines meet at right angles
- Bisect means to cut in half
- Two lines are perpendicular if the product of their gradients is -1
How do I find the equation of the perpendicular bisector of a line segment?
- To find the equation of a straight line you need to find
- The gradient of the line
- A coordinate of a point on the line
- To find the equation of the perpendicular bisector of a line segment follow these steps:
-
- STEP 1: Find the coordinates of the midpoint of the line segment
- We know that the perpendicular bisector will cut the line segment in half so we can use the midpoint of the line segment as the known coordinate on the bisector
- STEP 2: Find the gradient of the line segment
- STEP 1: Find the coordinates of the midpoint of the line segment
-
- STEP 3: Find the gradient of the perpendicular bisector
- This will be -1 divided by the gradient of the line segment
- STEP 4: Substitute the gradient of the perpendicular bisector and the coordinates of the midpoint into an equation for a straight line
- The point-gradient form is the easiest
- STEP 5: Rearrange into the required form
- Either or
- These equations for a straight line are given in the formula booklet
- STEP 3: Find the gradient of the perpendicular bisector
Worked example
Point A has coordinates (4, -6) and point B has coordinates (8, 6). Find the equation of the perpendicular bisector to [AB ]. Give your answer in the form .
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