Working with Coordinates in All Four Quadrants

The Coordinate Axes

The coordinate axes, $x$ and $y$, provide a systematic way of locating any point within a flat plane. These two lines intersect at a central point called the origin. The intersection of these axes divides the entire plane into four distinct regions known as quadrants.

In standard representation, the $x$ axis is the horizontal line. Numbers to the right of the origin are positive, while numbers to the left of the origin are negative. The $y$ axis is the vertical line. Numbers above the origin are positive, and numbers below the origin are negative.

Identifying Points in Four Quadrants

The position of any point is expressed as two numbers enclosed in brackets, separated by a comma. This is written as $(x, y)$, where the $x$ coordinate always precedes the $y$ coordinate.

In the diagram above, the four quadrants are illustrated with specific points:

1. Point A is $(4, 2)$, located in the first quadrant where both $x$ and $y$ are positive.
2. Point B is $(-4, 2)$, located in the second quadrant where $x$ is negative and $y$ is positive.
3. Point C is $(-2, -4)$, located in the third quadrant where both $x$ and $y$ are negative.
4. Point D is $(2, -4)$, located in the fourth quadrant where $x$ is positive and $y$ is negative.

Solving Geometric Problems with Coordinates

Coordinates allow for the construction and analysis of geometric shapes. By knowing the properties of a shape, such as a square, you can deduce missing coordinates by calculating lengths along the axes.

Worked Example: Finding Coordinates of a Square

A square $ABCD$ is drawn on a coordinate grid. You are given that $A$ is the point $(-6, 4)$ and $B$ is the point $(3, 4)$. It is also known that vertex $C$ lies in the fourth quadrant and vertex $D$ lies in the third quadrant. What are the coordinates of $C$ and $D$?

Step 1: Calculate the side length. Point $A$ is 6 units to the left of the $y$ axis and 4 units above the $x$ axis. Point $B$ is 3 units to the right of the $y$ axis and 4 units above the $x$ axis. Since both $A$ and $B$ have the same $y$ coordinate, the line segment $AB$ is horizontal. The distance between them is $3 - (-6) = 9$ units. Therefore, the side length of the square is 9 units.

Step 2: Locate vertex $C$. Since $C$ is in the fourth quadrant and forms a square with $AB$, it must be 9 units directly below point $B$. Its $x$ coordinate remains 3, but its $y$ coordinate becomes $4 - 9 = -5$. Thus, $C$ is at $(3, -5)$.

Step 3: Locate vertex $D$. Similarly, $D$ is in the third quadrant and must be 9 units directly below point $A$. Its $x$ coordinate remains $-6$, and its $y$ coordinate becomes $4 - 9 = -5$. Thus, $D$ is at $(-6, -5)$.

The final coordinates are $C(3, -5)$ and $D(-6, -5)$, as shown in the diagram above.