Bisectors of the Cartesian Plane
In the Cartesian plane, the bisectors are the two lines that pass through the origin and divide the plane into equal angles. Every point on a bisector is equidistant from the x-axis and the y-axis. This property can be written as $$ |x| = |y| $$
In other words, the x-coordinate and y-coordinate of any point on a bisector have the same absolute value.
The Cartesian plane contains two bisectors, each with its own equation.
- Bisector of the First and Third Quadrants
On this line, the coordinates (x, y) have the same sign and equal absolute values. Its equation is $$ y = x $$ Every point whose coordinates are equal lies on this bisector. Since there are infinitely many such points, the equation has infinitely many solutions.
- Bisector of the Second and Fourth Quadrants
On this line, the coordinates (x, y) have opposite signs but the same absolute value. Its equation is $$ y = -x $$ Every point that satisfies this equation lies on the bisector, so this equation also has infinitely many solutions.
Both bisectors can be described simultaneously by the relation
$$ |x| = |y| $$
This equation represents all points whose distance from the x-axis is equal to their distance from the y-axis.
The bisectors are among the most important lines in analytic geometry. They are often used to study symmetry, solve geometric problems, and analyze the relationships between points and the coordinate axes.
