8.1 Rectangular Coordinate System

Graphs drawn on a rectangular coordinate system, known as the Cartesian coordinate system (invented by René Descartes), help provide information in a visual form. Understanding the rectangular coordinate system is crucial to be able to read and draw graphs, which is essential in many branches of mathematics.

The rectangular coordinate system uses a horizontal and a vertical number line, each known as an axis. These two perpendicular axes cross at the point (O), known as the origin.

The horizontal number line (moving to the left or the right) is called the X-axis, and the vertical number line (moving up or down) is called the Y-axis, as illustrated in Exhibit 8.1-a.

The numbers to the right of the origin along the X-axis are positive ( + ), and those to the left are negative ( − ). The numbers above the origin along the Y-axis are positive ( + ), and those below are negative ( − ).

The purpose of the rectangular coordinate system and the sign convention is to locate a point relative to the X- and Y-axes and in reference to the origin ‘O’.

Points in the Rectangular Coordinate System

A point in the Cartesian coordinate system is a location in the plane, represented as an ordered pair of numbers inside a set of brackets called coordinates. The first number is called the x-coordinate, representing its horizontal position with respect to the origin, and the second is called the y-coordinate, representing its vertical position with respect to the origin. The ordered pair of coordinates for a given point P is written as P(x, y) or (x, y). For example, the origin (i.e., the point where the x-axis and y-axis intersect) is identified by the coordinates (0, 0) since its x and y coordinates are 0.

As illustrated in Exhibit 8.1-b, the ordered pair (2, 3) refers to the coordinates of point P, which is 2 units to the right and 3 units above, in reference to the origin.

It is important to identify the coordinate numbers in their order. They are called ordered pairs because the order in which they appear determines their position on the graph. Changing the order of the coordinates will result in a different point.

For example, (2, 3) and (3, 2) are different points.

• (2, 3) refers to a point ‘P’, 2 units to the right of the origin and 3 units above the origin.
• (3, 2) refers to a point ‘Q’, 3 units to the right of the origin and 2 units above the origin.

It is called a rectangular coordinate system because the x- and y-coordinates form a rectangle with the X- and Y-axes, as seen in the exhibit above.

Quadrants

The X- and Y-axes divide the coordinate plane into four regions called quadrants. Quadrants are numbered counter-clockwise from one (I) to four (IV), starting from the upper-right quadrant, as illustrated in Exhibit 8.1-c.

The upper-right quadrant is Quadrant I, the upper-left quadrant is Quadrant II, the lower-left quadrant is Quadrant III, and the lower-right quadrant is Quadrant IV. Table 8.1 shows the sign convention of coordinates in each quadrant, with examples plotted on the graph in Exhibit 8.1-d.

Table 8.1: Sign Convention of Coordinates in Different Quadrants, Axes, and Origin

Solution

A: (6, 3) 6 units along the x-axis to the right, 3 units up the y-axis.
B: (−2, 5) 2 units to the left along the x-axis, 5 units up the y-axis.
C: (−7, −3) 7 units to the left along the x-axis, 3 units down the y-axis.
D: (4, −6) 4 units to the right along the x-axis, 6 units down the y-axis.
E: (0, 4) 0 units along the x-axis, 4 units up the y-axis.
F: (−5, 0) 5 units to the right along the x-axis, 0 units up or down the y-axis.
G: (0, −6) 0 units along the x-axis, 6 units down the y-axis.
H: (3, 0) 3 units to the left along the x-axis, 0 units up or down the y-axis.

Solution

1. A (−15, 20) [latex]\longrightarrow[/latex] (−, +) = 2^nd Quadrant
2. B (20, 5) [latex]\longrightarrow[/latex] (+, +) = 1^st Quadrant
3. C (9, 0) [latex]\longrightarrow[/latex] (+, 0) = X-Axis (Right)
4. D (0, 20) [latex]\longrightarrow[/latex] (0, +) = Y-Axis (Up)
5. E (12, −18) [latex]\longrightarrow[/latex] (+, −) = 4^th Quadrant
6. F (0, −6) [latex]\longrightarrow[/latex] (0, −) = Y-Axis (Down)
7. G (−30, −15) [latex]\longrightarrow[/latex] (−, −) = 3^rd Quadrant
8. H (−1, 0) [latex]\longrightarrow[/latex] (−, 0) = X-Axis (Left)

Solution

Plotting points A, B, and C:

A (–3, 3): 3 units to the left of the origin and 3 units above the origin

B (4, 3): 4 units to the right of the origin and 3 units above the origin

C (4, –2): 4 units to the right of the origin and 2 units below the origin

Connecting point A to point B results in a horizontal line (since they share the same y-coordinate), and connecting point B to point C results in a vertical line (since they share the same x-coordinate).

The 4^th vertex of the rectangle, D, will have the same x-coordinate as point A and the same y-coordinate as point C.

Therefore, the coordinates for the 4^th vertex are D (–3, –2).

Solution

Plotting point P:

P (–2, 1): 2 units to the left of the origin and 1 unit above the origin

Since we are drawing a vertical line, point Q will have the same x-coordinate as point P.

One possible set of coordinates for the other end of the line is 3 units above point P, i.e., Q (–2, 4).

The other possible set of coordinates for the other end of the line is 3 units below point P, i.e., Q (–2, –2).

8.1 Exercises

Answers to the odd-numbered questions are available at the end of the book.

For problems 1 to 4, plot the points on a graph.

1. a. A (−3, 5)
   b. B (5, −3)
   c. C (0, −4)
2. a. A (−6, 0)
   b. B (4, −2)
   c. C (0, −7)
3. a. D (6, 0)
   b. E (−2, 4)
   c. F (5, 2)
4. a. D (8, 0)
   b. E (−3, −5)
   c. F (5, 5)

For problems 5 to 8, determine the quadrant or axis in which the points lie.

5. a. A (−1, 2)
   b. B (5, −1)
   c. C (3, 5)
6. a. A (1, 6)
   b. B (4, −3)
   c. C (−7, 3)
7. a. D (−4, 0)
   b. E (−2, −7)
   c. F (0, 5)
8. a. D (6, 0)
   b. E (−1, −13)
   c. F (0, −7)

For problems 9 to 12, plot the pairs of points on a graph and calculate the length of each horizontal line joining the pair of points.

9. a. (3, 4) and (5, 4)
   b. (−7, 1) and (2, 1)
10. a. (2, –6) and (7, –6)
    b. (–5, –4) and (0, –4)
11. a. (−5, 3) and (0, 3)
    b. (−2, −2) and (6, −2)
12. a. (–6, 8) and (–1, 8)
    b. (7, –5) and (2, –5)

For problems 13 to 16, plot the pairs of points on a graph and calculate the length of each vertical line joining the pair of points.

13. a. (3, 6) and (3, 1)
    b. (5, 2) and (5, –5)
14. a. (–3, –5) and (–3, –9)
    b. (–3, 0) and (–3, 6)
15. a. (5, 6) and (5, 2)
    b. (7, 2) and (7, −4)
16. a. (−3, 5) and (−3, −4)
    b. (−3, 5) and (−3, 0)

17. Three vertices of a square ABCD have points A (−3, 3), B (1, 3), and C (1, −1). Find the coordinates of the 4^th vertex D.
18. Three vertices of a square EFGH have points E (–1, –2), F (6, –2), and G (6, 5). Find the coordinates of the 4^th vertex H.
19. Three vertices of a rectangle PQRS have points P (−3, 4), Q (6, 4), and R (6, −1). Find the coordinates of the 4^th vertex S.
20. Three vertices of a rectangle TUVW have points T (–4, 7), U (5, 7), and V (5, 4). Find the coordinates of the 4^th vertex W.
21. A vertical line has a length of 7 units, and the coordinates of one end of the line are (1, 5). Find the possible coordinates of the other end of the line.
22. A vertical line has a length of 5 units, and the coordinates of one end of the line are (–3, 1). Find the possible coordinates of the other end of the line.
23. A horizontal line has a length of 6 units, and the coordinates of one end of the line are (−1, 3). Find the possible coordinates of the other end of the line.
24. A horizontal line has a length of 8 units, and the coordinates of one end of the line are (–1, –2). Find the possible coordinates of the other end of the line.