 JUNIOR MATHS

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Revision Sheets

## 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.