Key Notes for Chapter (1) : Introduction to Coordinate Geometry


1) The Midpoint Formula

If $M(x,y)$ is the midpoint between the two endpoints $A(x_1,y_1)$ and $B(x_2,y_2)$ then $$ \displaystyle M(x,y)=\left( {\frac{{{{x}_{1}}+{{x}_{2}}}}{2}+\frac{{{{y}_{1}}+{{y}_{2}}}}{2}} \right)$$

2) Distance Formula

The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is $$ \displaystyle \sqrt{{{{{\left( {{{x}_{2}}-x} \right)}}^{2}}+{{{\left( {{{y}_{2}}-{{y}_{1}}} \right)}}^{2}}}}$$

3) The slope of a Line

$$ \displaystyle \text{slope }(m)=\frac{{\text{vertical change}}}{{\text{horizontal change}}}=\frac{{\text{rise}}}{{\text{run}}}$$

4) Slope Formula

If the slope of the line passing through any two points $A(x_1, y_1)$ and $B(x_2, y_2)$ is $m$, then $$m=\displaystyle\frac{y_2-y_1}{x_2-x_1}$$

5) Positive Slope

A line ascending from left to right has a positive slope.

6) Negative Slope

A line descending from left to right has a negaitive slope.

7) Zero Slope

The slope of the horizontal line is zero.

8) Undefined Slope

The slope of the vertical line is undefined.

9) Slope-Intercept Form of a Line

The equation of the form
$$y=mx+c$$ is the equation of a straight line with slope $m$ and $y$-intercept $c$, which is called the slope-intercept form.

10) Point-Slope Form of a Line

The equation of a straight line, with slope $m$, and passes through the point $(x_1, y_1)$ is
$$y-y_1=m(x-x_1)$$ which is called the point-slope form of a straight line.

11) Horizontal Line

  • The $X$-axis and all lines parallel to it are called horizontal lines.
  • The equation of horizontal line intersecting the $Y$-axis at $(0,c)$ is $y=c$.
  • The equation of the $X$-axis is $y=0$.

12) Vertical Line

  • The $Y$-axis and all lines parallel to it are called vertical lines.
  • The equation of vertical line intersecting the $X$-axis at $(a,0)$ is $x=a$.
  • The equation of the $Y$-axis is $x=0$.

13) Parallel Lines and Perpendicular Lines

  • Any two horizontal lines are parallel.
  • Any two vertical lines are parallel.
  • Vertical and horizontal lines are perpendicular to each other.

14) Some Important Properties

  • Two non-vertical lines are parallel if and only if they have the same slope.
  • Two non-vertical lines are perpendicular if and only if the product of their slopes is -1 (i.e., one is the negative reciprocal of the other).
  • On a same straight line all segments have the same slope.
  • Three or more points that lie on the same straight line are said to be collinear.