Tenth Grade Mathematics: Review of the Cartesian Plane and Plotting
Coordinate geometry, also known as analytic geometry, is the mathematical discipline that establishes a one-to-one correspondence between geometric points in space and ordered algebraic pairs of real numbers. Conceived by René Descartes, this unified perspective allows geometric theorems about collinearity, concurrency, and congruence to be verified with algebraic calculations, and conversely allows abstract algebraic relations to be visualized as geometric shapes. Before developing the advanced metric formulas of tenth-grade coordinate geometry, we must review the foundational anatomy and sign conventions of the Cartesian coordinate plane.
The Structure of the Coordinate Plane
The two-dimensional Cartesian plane is constructed by placing two real number lines perpendicular to each other so that they intersect at their mutual zero points:
y-axis
|
4 +
|
QUADRANT II 3 + QUADRANT I
(-, +) [Abscissa < 0, | (+, +) [Abscissa > 0,
Ordinate > 0] 2 + Ordinate > 0]
|
1 +
|
---+-----+-----+-----+-----+---+---+-----+-----+-----+-----+-----+--- x-axis
-5 -4 -3 -2 -1 0 1 2 3 4 5
| Origin O(0, 0)
QUADRANT III -1 + QUADRANT IV
(-, -) [Abscissa < 0, | (+, -) [Abscissa > 0,
Ordinate < 0] -2 + Ordinate < 0]
|
-3 +
|
-4 +
|
The horizontal line is the x-axis, and the vertical line is the y-axis. Their unique intersection point is the origin O, represented by the ordered pair (0, 0).
Terminology of Coordinates
For any point P located in the plane:
Definitions:
1. Ordered Pair: P(x, y)
2. Abscissa (x-coordinate): The perpendicular distance of the point P from the
y-axis, measured with sign along the horizontal x-axis.
3. Ordinate (y-coordinate): The perpendicular distance of the point P from the
x-axis, measured with sign along the vertical y-axis.
Special Loci on the Axes:
- Every point lying directly on the x-axis has its ordinate y = 0: (x, 0).
- Every point lying directly on the y-axis has its abscissa x = 0: (0, y).
- The origin has both coordinates zero: (0, 0).
Sign Conventions Across Quadrants
The intersecting perpendicular axes partition the infinite plane into four open angular regions called quadrants, numbered counterclockwise starting from the upper-right:
+--------------+------------------+------------------+-----------------------+
| Quadrant | Sign of Abscissa | Sign of Ordinate | Standard Ordered Pair |
+--------------+------------------+------------------+-----------------------+
| Quadrant I | Positive (x > 0) | Positive (y > 0) | (+, +) |
+--------------+------------------+------------------+-----------------------+
| Quadrant II | Negative (x < 0) | Positive (y > 0) | (-, +) |
+--------------+------------------+------------------+-----------------------+
| Quadrant III | Negative (x < 0) | Negative (y < 0) | (-, -) |
+--------------+------------------+------------------+-----------------------+
| Quadrant IV | Positive (x > 0) | Negative (y < 0) | (+, -) |
+--------------+------------------+------------------+-----------------------+
Points lying directly on the axes do not belong to any quadrant; they form the boundaries separating the quadrants.
Reflections Across Axes and the Origin
Coordinate geometry allows geometric reflections (transformations across lines of symmetry) to be computed with basic sign inversions:
Rules for Coordinate Reflections:
1. Reflection across the x-axis:
Inverts the sign of the y-coordinate while keeping x unchanged:
(x, y) ===> (x, -y)
2. Reflection across the y-axis:
Inverts the sign of the x-coordinate while keeping y unchanged:
(x, y) ===> (-x, y)
3. Reflection through the Origin (Point Symmetry):
Inverts the signs of BOTH coordinates:
(x, y) ===> (-x, -y)
Chapter Exercises
Exercise 1: State the quadrant or axis on which each of the following points lies: A(3, -5), B(-4, -7), C(0, 6), D(-8, 2), E(5, 0), and F(0, 0).
Exercise 2: Write down the coordinates of the image of the point P(-3, 4) when reflected: (a) in the x-axis, (b) in the y-axis, and (c) in the origin.
Exercise 3: Three vertices of a rectangle ABCD are A(2, 3), B(-3, 3), and C(-3, -2). Plot these points conceptually and determine the coordinates of the fourth vertex D, and find the perimeter of the rectangle.
Solutions and Explanations
Solution 1: Point A(3, -5) has a positive x-coordinate and a negative y-coordinate, so it lies in Quadrant IV. Point B(-4, -7) has both coordinates negative, placing it in Quadrant III. Point C(0, 6) has an x-coordinate of 0 and a non-zero y-coordinate, meaning it lies directly on the y-axis. Point D(-8, 2) has a negative x-coordinate and a positive y-coordinate, so it lies in Quadrant II. Point E(5, 0) has a y-coordinate of 0, meaning it lies directly on the x-axis. Point F(0, 0) is the origin, which lies at the intersection of both axes.
Solution 2: For point P(-3, 4): For part (a), reflection in the x-axis leaves the x-coordinate unchanged and reverses the sign of the y-coordinate, giving P1(-3, -4). For part (b), reflection in the y-axis leaves the y-coordinate unchanged and reverses the sign of the x-coordinate, giving P2(3, 4). For part (c), reflection in the origin reverses the signs of both coordinates, giving P3(3, -4).
Solution 3: In a rectangle ABCD, opposite sides are equal in length and parallel to the axes. Side AB joins A(2, 3) and B(-3, 3); since both points share y = 3, AB is a horizontal segment with length |2 - (-3)| = 5 units. Side BC joins B(-3, 3) and C(-3, -2); since both points share x = -3, BC is a vertical segment with length |3 - (-2)| = 5 units. For vertex D to complete rectangle ABCD, side CD must be horizontal from C(-3, -2), having y = -2, and side AD must be vertical from A(2, 3), having x = 2. Therefore, the coordinates of the fourth vertex are D(2, -2). The side lengths are length = 5 units and width = 5 units (making this rectangle a square). The perimeter is 4 * 5 = 20 units.