Coordinate Geometry

The One Thing

Coordinate geometry describes straight lines by linking coordinates, graphs, equations, and geometric relationships. The gradient and y-intercept determine a non-vertical straight line in the form $y=mx+c$, while algebraic methods can be used to investigate parallelism, perpendicularity, midpoints, distances, and intersections.

Definitions and Results

• Cartesian coordinate system: A grid formed by a horizontal $x$-axis and a vertical $y$-axis, used to locate points written as $(x,y)$.
• Origin: The point where the axes meet, with coordinates $(0,0)$.
• Gradient: The change in $y$ divided by the change in $x$, measuring the steepness and direction of a line. For points $(x_1,y_1)$ and $(x_2,y_2)$,

$$m=\frac{y_2-y_1}{x_2-x_1},$$ provided $x_2\ne x_1$. The changes in $y$ and $x$ must be taken in the same order.

• Rise over run: The calculation of gradient as vertical change divided by horizontal change.
• Positive gradient: A line rises from left to right.
• Negative gradient: A line falls from left to right.
• Horizontal line: A line with gradient $0$, written as $y=a$, where $a$ is constant.
• Vertical line: A line with undefined gradient, written as $x=a$, where $a$ is constant.
• $y$-intercept: The value of $y$ where a line crosses the $y$-axis; it is found when $x=0$.
• $x$-intercept: The value of $x$ where a line crosses the $x$-axis; it is found when $y=0$.
• Equation of a straight line: A rule connecting the $x$- and $y$-values of every point on a line. A common form is

$$y=mx+c.$$

• Gradient-intercept form: In $y=mx+c$, $m$ is the gradient and $c$ is the $y$-intercept. These values completely determine a non-vertical straight line.
• Alternative line forms: A line may also be written as

$$ax+by+c=0$$ or $$y-y_1=m(x-x_1).$$

• Line segment: The part of a straight line between two given endpoints.
• Parallel lines: Lines that never meet and have equal gradients.
• Perpendicular lines: Lines that meet at a right angle. For two non-vertical lines,

$$m_1\times m_2=-1.$$

• Midpoint: The point exactly halfway between two endpoints. The midpoint of $(x_1,y_1)$ and $(x_2,y_2)$ is

$$\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right).$$

• Distance between two points: The length of the line segment joining two points. The distance between $(x_1,y_1)$ and $(x_2,y_2)$ is

$$\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2},$$ using the Pythagorean theorem.

• Point-on-line test: Substitute a point’s coordinates into the line equation. A true statement confirms that the point lies on the line.
• Intersection of two lines: The point common to both lines, found by solving their simultaneous equations.
• Line graph requirements: Axes must be clearly labelled, the scale must be suitable, points must be plotted accurately, and a straight line must be drawn through the data or points.

Worked Methods

Finding the gradient between two points

• Identify the points $(x_1,y_1)$ and $(x_2,y_2)$.
• Calculate the change in $y$: $y_2-y_1$.
• Calculate the change in $x$: $x_2-x_1$.
• Divide the change in $y$ by the change in $x$:

$$m=\frac{y_2-y_1}{x_2-x_1}.$$

• Keep the order consistent in both numerator and denominator to avoid sign errors.
• If $x_2=x_1$, the line is vertical and its gradient is undefined.

Finding the equation of a straight line

• Calculate the gradient $m$.
• Write the equation in gradient-intercept form:

$$y=mx+c.$$

• Substitute a known point $(x_1,y_1)$ into the equation:

$$y_1=m{x}_1+c.$$

• Rearrange to find $c$.
• State the completed equation.
• Check the equation by substituting the known point or another relevant coordinate.

Finding intercepts

• For the $y$-intercept, set $x=0$ in the line equation and solve for $y$.
• For the $x$-intercept, set $y=0$ and solve for $x$.
• Express each intercept as a coordinate if required.

Testing whether a point lies on a line

• Take the coordinates of the point.
• Substitute them for $x$ and $y$ in the line equation.
• Simplify both sides.
• If the resulting statement is true, the point lies on the line; otherwise, it does not.

Finding parallel lines

• Determine the gradient of the given line.
• Use the same gradient for the parallel line:

$$m_{\text{parallel}}=m_{\text{given}}.$$

• Substitute a known point on the required line into $y=mx+c$.
• Find $c$ and state the equation.
• Confirm parallelism by comparing the gradients.

Finding perpendicular lines

• Determine the gradient $m_1$ of the first non-vertical line.
• Use the condition

$$m_1{m}_2=-1.$$

• Solve for the perpendicular gradient $m_2$:

$$m_2=-\frac{1}{m_1}.$$

• Substitute a known point on the perpendicular line into $y=mx+c$.
• Find $c$ and state the equation.

Finding the midpoint

For endpoints $(x_1,y_1)$ and $(x_2,y_2)$:

• Add the two $x$-coordinates and divide by $2$.
• Add the two $y$-coordinates and divide by $2$.
• Write the result as

$$\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right).$$

Finding the distance between two points

For $(x_1,y_1)$ and $(x_2,y_2)$:

• Calculate the horizontal change $x_2-x_1$.
• Calculate the vertical change $y_2-y_1$.
• Square both changes.
• Add the squares.
• Take the square root:

$$d=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}.$$

Finding the intersection of two lines

• Write the two line equations.
• Set the expressions equal if both are written in terms of $y$.
• Solve the simultaneous equations for $x$.
• Substitute the value of $x$ into either equation to find $y$.
• State the intersection as an ordered pair and check it in both equations.

Drawing a line graph

• Draw the horizontal $x$-axis and vertical $y$-axis.
• Label both axes clearly.
• Select a suitable, consistent scale.
• Plot the coordinates accurately.
• Draw a straight line through the data or plotted points.
• Check that the graph is consistent with the calculated gradient and equation.

Where It Goes Wrong

• Using changes in $y$ and $x$ in different orders in

$$\frac{y_2-y_1}{x_2-x_1}$$ produces an incorrect sign.

• Forgetting that a vertical line has undefined gradient, rather than gradient $0$; its equation is $x=a$.
• Confusing the $x$-intercept, found when $y=0$, with the $y$-intercept, found when $x=0$.
• In $y=mx+c$, identifying $m$ as the gradient and $c$ as the $y$-intercept is essential.
• Applying $m_1{m}_2=-1$ to a vertical line is invalid because the condition applies to non-vertical lines.
• Failing to check a calculated equation, midpoint, intersection, or point by substitution or comparison with the graph can leave an algebraic error undetected.

What Gets Asked

• Calculate the gradient of a line from two coordinates or from a graph using rise over run.
• Interpret whether a line has a positive, negative, zero, or undefined gradient.
• Find the equation of a line from its gradient and a known point.
• Identify or calculate the $x$- and $y$-intercepts.
• Rewrite a straight-line equation in gradient-intercept form or another stated form.
• Test whether a given point lies on a line by substitution.
• Determine whether two lines are parallel or perpendicular.
• Find the equation of a line parallel or perpendicular to a given line through a specified point.
• Calculate the midpoint or distance between two points.
• Find the intersection of two straight lines by solving simultaneous equations.
• Draw and interpret a line graph using labelled axes, a suitable scale, accurately plotted points, and a straight line through the data or points.