Important Formulas: Orienting Yourself: The Use of Coordinates

1. Coordinates of a Point

For a point P in the Cartesian plane, its coordinates are written as:

$P(x,y)$    where    $x$ = abscissa (x-coordinate)    and    $y$ = ordinate (y-coordinate)

• x-coordinate (abscissa): Perpendicular distance of P from the y-axis, measured along the x-axis.
  $x=\text{distance from y-axis}$
• y-coordinate (ordinate): Perpendicular distance of P from the x-axis, measured along the y-axis.
  $y=\text{distance from x-axis}$

1.2 Special Coordinate Cases

• Point on the x-axis: $(x,0)$
• Point on the y-axis: $(0,y)$
• Origin O: $(0,0)$
• The x-coordinate of any point on the y-axis is 0.
• The y-coordinate of any point on the x-axis is 0.

2. Ordered Pair Rule

If  $x=y$,   then  $(x,y)=(y,x)$
If  $x\ne y$,   then  $(x,y)\ne (y,x)$

The order of coordinates matters. Interchanging the x and y values gives a different point unless both coordinates are equal.

3. Quadrant Sign Convention

The coordinate axes divide the Cartesian plane into four quadrants, numbered I to IV (anti-clockwise from the top-right):

Quadrant	x-coordinate	y-coordinate	Sign Form
I	Positive (+)	Positive (+)	$(+,+)$
II	Negative (-)	Positive (+)	$(-,+)$
III	Negative (-)	Negative (-)	$(-,-)$
IV	Positive (+)	Negative (-)	$(+,-)$

4. Distance Formula

The distance between two points $A(x_1,y_1)$ and $B(x_2,y_2)$ in the 2-D plane is given by the Baudhāyana-Pythagoras Theorem:

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

The sign of $(x_2-x_1)$ or $(y_2-y_1)$ does not affect the result since both differences are squared.

Example: Distance between $A(2,3)$ and $B(5,7)$:

$$AB=\sqrt{(5-2{)}^2+(7-3{)}^2}=\sqrt{3^2+4^2}=\sqrt{9+16}=\sqrt{25}=5\text{ units}$$

4.1 Special Cases of Distance Formula

• Distance between two points with the same y-coordinate(points on a horizontal line):$$d=\mathrm{\mid }{x}_2-x_1\mathrm{\mid }$$
• Distance between two points with the same x-coordinate(points on a vertical line):$$d=\mathrm{\mid }{y}_2-y_1\mathrm{\mid }$$
• Distance of a point $P(x,y)$ from the origin$O(0,0)$:$$OP=\sqrt{x^2+y^2}$$

5. Distance of a Point from the Axes

• Distance of $P(x,y)$ from the x-axis:$\mathrm{\mid }y\mathrm{\mid }$
• Distance of $P(x,y)$ from the y-axis:$\mathrm{\mid }x\mathrm{\mid }$

Example: For point C(-3, -2): Distance from x-axis = $\mathrm{\mid }-2\mathrm{\mid }$ = 2 units;  Distance from y-axis = $\mathrm{\mid }-3\mathrm{\mid }$ = 3 units.

6. Reflection in an Axis

Reflection in an axis preserves all side lengths of a figure.

• Reflection of $P(x,y)$ in the y-axis:$P^{\prime }(-x,y)$
• Reflection of $P(x,y)$ in the x-axis:$P^{\prime }(x,-y)$

If triangle PQR reflects to triangle P′Q′R′, then all corresponding side lengths are equal:

$$PQ=P^{\prime }{Q}^{\prime },QR=Q^{\prime }{R}^{\prime },PR=P^{\prime }{R}^{\prime }$$
Because squaring eliminates sign differences, the distance formula gives the same result for a figure and its reflection.