COORDINATE GEOMETRY
OVERVIEW
Below, we have provided our revision sheet for this topic, split into the following sections:
1) KEY POINTS
A summary of the most important aspects of the topic.
2) CAUSES OF CONFUSION
A list of mistakes that are commonly made by students.
3) EXAM TIPS
A collection of tips and tricks to give students that extra edge in their exam.
1) Key Points
The slope \(m\) of a line containing the points \((x_1,y_1)\) and \((x_2,y_2)\) is given by:
\begin{align}m=\frac{y_2-y_1}{x_2-x_1}\end{align}
The length \(l\) of a line segment whose endpoints are \((x_1,y_1)\) and \((x_2,y_2)\) is given by:
\begin{align}l=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}\end{align}
The midpoint of a line segment whose endpoints are \((x_1,y_1)\) and \((x_2,y_2)\) is given by:
\begin{align}\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)\end{align}
The equation of a line is of the form:
\begin{align}y=mx+c\end{align}
where \(m\) is the slope (steepness) of the line and \((0,c)\) is the point on the line where it crosses the \(y\) axis.
To find the equation of a line with slope \(m\) and containing the point \((x_1,y_1)\), we use the following formula:
\begin{align}y-y_1=m(x-x_1)\end{align}
To find the equation of a line containing the points \((x_1,y_1)\) and \((x_2,y_2)\), we first use the slope formula:
\begin{align}m=\frac{y_2-y_1}{x_2-x_1}\end{align}
and then use the following formula:
\begin{align}y-y_1=m(x-x_1)\end{align}
If a line of slope \(m_1\) is parallel to a line of slope \(m_2\), then \(m_1=m_2\).
If a line of slope \(m_1\) is perpendicular to a line of slope \(m_2\), then \(m_1m_2=-1\).
To find the point at which two lines intersect, we solve the equations of the lines simultaneously.
2) Causes of Confusion
In the Formulae & Tables booklet, both of the following are stated as the “equation of a line”:
\begin{align}y-y_1=m(x-x_1)\end{align}
\begin{align}y=mx+c\end{align}
In reality, the first equation is instead a formula that can be used to obtain the equation of a line (the second equation).
One of the most common mistakes that students mistake is to assume that the two points that they are given are \((x_1,x_2)\) and \(y_1,y_2)\).
Recall instead that the two points are \((x_1,y_1)\) and \(x_2,y_2)\)!
3) Exam Tips
When answering coordinate geometry questions, the difficulty is often in figuring out what formula(e) you need rather than actually using that formula(e).
We therefore recommend taking a minute or two ensuring that you are indeed using the right formula(e) before you being writing.
