Coordinate Geometry

The One Thing

Coordinate geometry converts geometric relationships into algebraic conditions involving coordinates, formulas, and equations. The principal methods determine distance, division, slope, collinearity, area, line equations, intersections, and loci, with applications to location and measurement problems.

Definitions and Results

• Cartesian Plane: A plane formed by two perpendicular number lines, the x-axis and y-axis, meeting at the origin.
• Coordinates of a Point: An ordered pair $(x,y)$, where $x$ is the abscissa and $y$ is the ordinate.
• Quadrants: The four regions formed by the coordinate axes: I $(+,+)$, II $(-,+)$, III $(-,-)$, and IV $(+,-)$.
• Distance Formula: For $(x_1,y_1)$ and $(x_2,y_2)$,

$$d=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}.$$ It follows from the Pythagorean theorem.

• Section Formula: If a point divides the segment joining $(x_1,y_1)$ and $(x_2,y_2)$ internally in the ratio $m:n$, its coordinates are

$$\left(\frac{m{x}_2+n{x}_1}{m+n},\frac{m{y}_2+n{y}_1}{m+n}\right).$$ For external division, $$\left(\frac{m{x}_2-n{x}_1}{m-n},\frac{m{y}_2-n{y}_1}{m-n}\right),$$ provided $m\ne n$.

• Midpoint: The point dividing a segment into two equal parts:

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

• Slope or Gradient: For two points,

$$m=\frac{y_2-y_1}{x_2-x_1},$$ when $x_2\ne x_1$. A vertical line has undefined slope; a horizontal line has slope zero.

• Collinearity: Three or more points are collinear if they lie on one straight line. This may be tested by equal slopes or by showing that the area formed by the points is zero.
• Area of a Triangle: For vertices $(x_1,y_1)$, $(x_2,y_2)$, and $(x_3,y_3)$,

$$\text{Area}=\frac{1}{2}\left|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)\right|.$$

• Straight Line: A locus of points satisfying a first-degree equation in $x$ and $y$.
• Slope-Intercept Form:

$$y=mx+c,$$ where $m$ is the slope and $c$ is the y-intercept.

• Point-Slope Form: A line of slope $m$ through $(x_1,y_1)$ has equation

$$y-y_1=m(x-x_1).$$

• Two-Point Form: The line through $(x_1,y_1)$ and $(x_2,y_2)$ is

$$\frac{y-y_1}{y_2-y_1}=\frac{x-x_1}{x_2-x_1},$$ when the denominators are nonzero. Equivalently, $$y-y_1=\frac{y_2-y_1}{x_2-x_1}(x-x_1),$$ provided $x_1\ne x_2$.

• Intercept Form: A line with x-intercept $a$ and y-intercept $b$ has equation

$$\frac{x}{a}+\frac{y}{b}=1,$$ where $a$ and $b$ are nonzero.

• General Form of a Line:

$$ax+by+c=0,$$ where $a,b,c$ are constants and $a$ and $b$ are not both zero.

• Parallel Lines: Two non-vertical lines are parallel when their slopes are equal. For

$$a_{1}x+b_{1}y+c_1=0,a_{2}x+b_{2}y+c_2=0,$$ the condition is $$a_1{b}_2=a_2{b}_1.$$

• Perpendicular Lines: Two non-vertical lines are perpendicular when the product of their slopes is $-1$. In general form, the condition is

$$a_1{a}_2+b_1{b}_2=0.$$

• Angle Between Two Lines: For slopes $m_1$ and $m_2$,

$$\tan \theta =\left|\frac{m_2-m_1}{1+m_1{m}_2}\right|,$$ when the denominator is nonzero.

• Distance of a Point from a Line: The perpendicular distance from $(x_1,y_1)$ to $ax+by+c=0$ is

$$\frac{|a{x}_1+b{y}_1+c|}{\sqrt{a^2+b^2}}.$$

• Distance Between Parallel Lines: For

$$ax+by+c_1=0,ax+by+c_2=0,$$ the distance is $$\frac{|c_1-c_2|}{\sqrt{a^2+b^2}}.$$

• Intersection of Lines: The common point of two lines, found by solving their equations simultaneously.
• Locus: The set of all points satisfying a particular geometric condition, represented by an equation.

Additional standard line results are: $$y=0$$ for the x-axis, $$x=0$$ for the y-axis, $$y=k$$ for a line parallel to the x-axis, and $$x=k$$ for a line parallel to the y-axis.

For $ax+by+c=0$, the slope is $-a/b$ when $b\ne 0$, the x-intercept is $-c/a$ when $a\ne 0$, and the y-intercept is $-c/b$ when $b\ne 0$.

Worked Methods

Finding the distance between two points

• Identify the points $(x_1,y_1)$ and $(x_2,y_2)$.
• Calculate the horizontal and vertical differences, $x_2-x_1$ and $y_2-y_1$.
• Substitute into

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

• Interpret the result as the length of the segment joining the two points.

The formula treats the horizontal and vertical differences as the legs of a right triangle and applies the Pythagorean theorem.

Finding a midpoint

• Add the two x-coordinates and divide by $2$.
• Add the two y-coordinates and divide by $2$.
• State the midpoint as

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

Dividing a line segment in a ratio

• Identify the endpoints $(x_1,y_1)$, $(x_2,y_2)$ and the ratio $m:n$.
• For internal division, substitute into

$$\left(\frac{m{x}_2+n{x}_1}{m+n},\frac{m{y}_2+n{y}_1}{m+n}\right).$$

• For external division, substitute into

$$\left(\frac{m{x}_2-n{x}_1}{m-n},\frac{m{y}_2-n{y}_1}{m-n}\right),$$ ensuring that $m\ne n$.

• Check that the resulting point satisfies the stated type of division.

Finding the slope of a line

• Select two points on the line.
• Substitute their coordinates into

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

• If $x_2=x_1$, the line is vertical and its slope is undefined.
• If $y_2=y_1$, the line is horizontal and its slope is zero.

Testing collinearity

• Calculate the slopes of two pairs of the given points.
• If the slopes are equal, the points are collinear, provided the slopes are defined.
• Alternatively, calculate

$$\frac{1}{2}\left|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)\right|.$$

• The points are collinear if this area is zero.

Finding the area of a triangle

• Label the vertices $(x_1,y_1)$, $(x_2,y_2)$, and $(x_3,y_3)$.
• Substitute into

$$\text{Area}=\frac{1}{2}\left|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)\right|.$$

• Simplify the expression.
• Take the absolute value so that the area is non-negative.

Forming the equation of a straight line

• Determine which information is given.
• Use slope-intercept form $y=mx+c$ when the slope and y-intercept are known.
• Use point-slope form

$$y-y_1=m(x-x_1)$$ when a point and slope are known.

• Use two-point form when two points are known:

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

• Use intercept form

$$\frac{x}{a}+\frac{y}{b}=1$$ when the x- and y-intercepts are known.

• Convert to general form $ax+by+c=0$ when required.

Finding a parallel or perpendicular line

• Determine the slope or coefficients of the given line.
• For a parallel line, retain the same slope. In general form, a line through a given point parallel to

$$ax+by+c=0$$ has the form $$ax+by+k=0.$$

• Substitute the given point to determine $k$.
• For a perpendicular line, use the negative reciprocal of the original slope when the slopes are defined.
• Alternatively, use

$$a_1{a}_2+b_1{b}_2=0$$ in general form.

Finding the angle between two lines

• Determine the slopes $m_1$ and $m_2$.
• Substitute into

$$\tan \theta =\left|\frac{m_2-m_1}{1+m_1{m}_2}\right|.$$

• Ensure that $1+m_1{m}_2\ne 0$.
• Determine the required angle from the resulting tangent value.

Finding the distance from a point to a line

• Express the line as $ax+by+c=0$.
• Identify the point $(x_1,y_1)$.
• Substitute into

$$d=\frac{|a{x}_1+b{y}_1+c|}{\sqrt{a^2+b^2}}.$$

• Simplify and interpret the result as the perpendicular distance.

Finding the intersection of two lines

• Write both line equations.
• Solve them simultaneously using substitution or elimination.
• State the common ordered pair as the intersection point.
• Substitute the result into both original equations to check it.

Solving coordinate geometry problems systematically

• Draw a diagram where appropriate.
• Assign coordinates carefully.
• Translate the geometric statement into an algebraic condition.
• Select an appropriate formula or line equation.
• Solve accurately.
• Check the result in the original geometric condition.
• Interpret the algebraic result geometrically.

Coordinate geometry therefore supports applications in navigation, mapping, architecture, surveying, computer graphics, transportation planning, and data visualization.

Where It Goes Wrong

• The signs of coordinates are assigned incorrectly, especially when identifying Quadrants I $(+,+)$, II $(-,+)$, III $(-,-)$, and IV $(+,-)$.
• The distance formula is applied without preserving the coordinate differences or their squares:

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

• The slope formula is used for a vertical line even though $x_2-x_1=0$; a vertical line has undefined slope, whereas a horizontal line has slope zero.
• Internal and external section formulas are confused, or the condition $m\ne n$ is forgotten for external division.
• The absolute value is omitted in the triangle-area formula or the point-to-line distance formula, producing a negative “area” or distance.
• Conditions attached to line forms are ignored: $a$ and $b$ must be nonzero in intercept form, $x_1\ne x_2$ is required for the two-point slope expression, and $1+m_1{m}_2\ne 0$ is required for the stated angle formula.

What Gets Asked

This material supports questions requiring students to:

• Identify coordinates, abscissas, ordinates, axes, quadrants, and the origin.
• Calculate distances, midpoints, and internal or external division points.
• Find or compare slopes, including horizontal and vertical lines.
• Test whether points are collinear using equal slopes or zero area.
• Calculate the area of a triangle from its coordinate vertices.
• Form equations of lines using slope-intercept, point-slope, two-point, intercept, or general form.
• Determine slopes, intercepts, and equations from $ax+by+c=0$.
• Find equations of parallel and perpendicular lines.
• Calculate the angle between two lines.
• Find the distance from a point to a line and between parallel lines.
• Solve simultaneous line equations to determine intersections.
• Construct or interpret a locus represented by an equation.
• Translate practical situations involving navigation, mapping, architecture, surveying, computer graphics, transportation planning, or data visualization into coordinate methods.