The Cartesian plane , a refresher

Before we plunge into formulas, a brief look at the geometric stage.

Definitions

The Cartesian plane is the set of all ordered pairs $(x, y)$ of real numbers. Two perpendicular number lines , the horizontal $x$ -axis and the vertical $y$ -axis , meet at the origin $O = (0, 0)$ .

A point $P = (a, b)$ in the plane is plotted by moving $a$ units along the $x$ -axis (right if $a > 0$ , left if $a < 0$ ) and then $b$ units parallel to the $y$ -axis (up if $b > 0$ , down if $b < 0$ ).

The plane is divided into four quadrants by the axes:

Quadrant I: $x > 0, y > 0$ (top right).
Quadrant II: $x < 0, y > 0$ (top left).
Quadrant III: $x < 0, y < 0$ (bottom left).
Quadrant IV: $x > 0, y < 0$ (bottom right).

Points on the $x$ -axis have $y = 0$ ; on the $y$ -axis, $x = 0$ .

Quick facts

$P = (a, b)$ and $P' = (a, -b)$ are mirror images about the $x$ -axis.
$P = (a, b)$ and $P' = (-a, b)$ are mirror images about the $y$ -axis.
$P = (a, b)$ and $P' = (-a, -b)$ are reflections through the origin.
The distance of $P = (a, b)$ from the $x$ -axis is $|b|$ ; from the $y$ -axis is $|a|$ ; from the origin is $\sqrt{a^2 + b^2}$ (we'll derive this next).
The order $(x, y)$ matters: $(2, 5) \ne (5, 2)$ .

Concept: every point is a pair of numbers

The whole power of coordinate geometry lies in this: any geometric question about points can be translated into algebra by their coordinates, and any algebraic relation between coordinates becomes geometry. We will see this immediately in the next topic on distance.

For now, get comfortable with locating points. Plot a few:

$(3, 4)$ , Quadrant I.
$(-2, 5)$ , Quadrant II.
$(-3, -1)$ , Quadrant III.
$(4, -2)$ , Quadrant IV.
$(0, -3)$ , on the negative $y$ -axis.
$(7, 0)$ , on the positive $x$ -axis.

Worked examples

Example 1. Identify the quadrant of $(-5, 7)$ and the quadrant of $(2, -3)$ .

$(-5, 7)$ : $x < 0, y > 0$ , Quadrant II. $(2, -3)$ : $x > 0, y < 0$ , Quadrant IV.

Example 2. Find the coordinates of a point on the $x$ -axis at distance $4$ from the origin.

Such a point has $y = 0$ and $|x| = 4$ . So $(4, 0)$ or $(-4, 0)$ .

Example 3. The mirror image of $(3, -7)$ across the $y$ -axis is?

$y$ -axis reflection flips the sign of $x$ : $(-3, -7)$ .

Example 4. The point $(0, -4)$ lies on which axis, and at what distance from the origin?

On the negative $y$ -axis, distance $4$ from origin.

Example 5. What is the distance from the origin to the point $(3, 4)$ ?

By Pythagoras, $\sqrt{9 + 16} = 5$ . (Foreshadowing the distance formula!)

Try it yourself

Plot and identify the quadrant: $(2, 3), (-2, 3), (-2, -3), (2, -3), (0, 5), (-7, 0)$ .
A point lies on the $x$ -axis at $3$ units to the left of the origin. Coordinates?
Reflect $(5, -2)$ across the $x$ -axis.
Reflect $(5, -2)$ across the origin.
Distance of $(7, -24)$ from the origin?
The perpendicular distance from $(-3, 4)$ to the $y$ -axis is what?
Three points $(2, 0), (0, 3), (-5, 0)$ . Which are on the axes?
A point is equidistant from both axes, in Quadrant II, at perpendicular distance $5$ . Coordinates?
Origin's coordinates? What is its distance from any axis?
Where do points with $x = y$ lie?

Pitfalls / Insight

Order $(x, y)$ matters. $(3, 5)$ and $(5, 3)$ are different points.
Signs are key. A wrong sign on a coordinate plants the point in the wrong quadrant.
Distance from an axis equals the absolute value of the perpendicular coordinate.

Insight. The plane is a map. Every point has an address $(x, y)$ . The next two topics show how to compute distances between addresses and how to find addresses that divide a journey in a given ratio.