Coordinate Geometry

Coordinate Geometry involves the famous "Cartesian Coordinate Plane" by René Descartes which was developed on the 17th Century. It is a part of Geometry by which the points are located on the plane with the use of ordered pairs.

What is a COORDINATE PLANE?

A coordinate plane consists of a horizontal axis (x-axis) and a vertical axis (y-axis) which intersects at a common point called origin, perpendicularly. The x and y axis is formed by a ''number line, by which the numbers on it corresponds to a point in the line which is then called coordinate of a point.

Figure 1: This is an example of a coordinate plane.

• All points or values that go to the right of the x-axis (after the origin) are always positive.
• All the points or values that go to the left of the y-axis (after the origin) are always negative.
• All points or values that go above the y-axis (after the origin) are always positive.
• All points or values that go down the y-axis (after the origin) are always negative.

Ordered Pair

Ordered pairs are collectively called as "points" in the Coordinate plane. It consists of one coordinate point from the x-axis and one coordinate point from the y-axis, symbolized as (x,y). For example we have a point A with coordinates of (3,4) on the plane. Consider the picture below:

"3 is the x- coordinate, and 4 is the y-coordinate: collectively called as ordered pair"

- x- coordinate is also called Abscissa

-y-coordinate is also called Ordinate

Plotting points in the coordinate plane

It is important in Coordinate Geometry that you will learn how to locate and plot points. Remember: X coordinate comes first (always) before y-coordinate in the ordered pair. Also, remember to name the point on the plot (if a point name is indicated on the given coordinates)

To plot points:

Suppose, we are given point B (4,2):

" (1) locate first the x-coordinate (the blue arrow), (2) then locate the y-coordinate (pink arrow), (3) after which, you then plot by putting a point on the area which represents (4,2)."

Try to plot the following:

1. A ( 5,3)

2 B (3,2)

Lines in Coordinate Geometry

lines in Coordinate geometry are formed when two or more points are connected. Suppose we have A (5,4) and B (2,1). When these points in the plane are connected it will form a line, as shown on the image below:

"When points A and B were connected, it formed line AB."

Distance between Two points

The distance between two points is the length of the portion of the line/ line segment that connects one point to the other.

" the broken line is the distance between points X and Y."

The Distance Formula

From the image above we have points: X (-5,4) and Y (3,3). Let us find the distance between these two points.

Let (x₁, y₁) be (-5,4), and (x₂ , y₂) be (3,3). Substitute the following values to the formula:

Thus, the distance between point X and point Y is \sqrt{65}

Let us have another example. Suppose we have points E ( -2, -5) and F (4,3), let us find the distance between point E and F. Let (x₁, y₁) be (-2, -5) and (x₂ , y₂) be (4,3). We then substitute the following values to the formula:

Thus, the distance between E and F is 10.

Try to get the distance between 2 points on the following items:

1. A (-2, -3), B (2,4)

2. C (-3, -4), D (-2, 1)

3. E (-5,5), F (5, -5)

plot the following points, connect it (form a line) and get its distance:

4. G (-4,4). H (3, 5)

d		=		\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}
d		=		\sqrt{( 3-(-4) )^2+( 5 - 4 )^2}
d		=		\sqrt{(7)^2+(1)^2}
d		=		\sqrt{49+1}
d		=		\sqrt{50}

5. I (-3,1), J (2, -4)

d		=		\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}
d		=		\sqrt{( 2-(-3) )^2+( (-4) - 1)^2}
d		=		\sqrt{(5)^2+(-5)^2}
d		=		\sqrt{25 + 25}
d		=		\sqrt{50}

Midpoint Formula

A midpoint is that which is halfway between two points.

In finding the coordinates of the midpoint of a line segment, we use the formula:

For example: Find the coordinates of the midpoint of A (5,4) and B (2,1).

Let (x₁, y₁) be (5,4), (x₂, y₂) be (2,1).

Substitute the following values to the formula.

Therefore, the coordinates of the midpoint of line AB is (3.5, 2.5)

Another example:

Find the coordinates of the midpoint of Y (-4,3) and Z (3, -1).

Let (x₁, y₁) be (-4,3), (x₂, y₂) be (3,-1).

Substitute the following values to the formula.

Therefore the coordinates of the midpoint of line YZ is (-0.5, 1)

Try to answer the following:

Find the coordinates of the midpoint of:

1. (5,2), (3, -2)

Let (x₁, y₁) be (5,2), (x₂, y₂) be (3,-2).

Substitute the following values to the formula.

x		=	1\over{2}(x_1 +x_2)
x		=	1\over{2}(5 + 3)
x		=	1\over{2}(8)
x		=	4

y		=	1\over{2}(y_1 +y_2)
y		=	1\over{2}(2 + (-2))
y		=	1\over{2}(0)
y		=	0

Thus, the coordinates are (4,0)

2. (2, 4), (3,0)

Let (x₁, y₁) be (2,4), (x₂, y₂) be (3,0).

Substitute the following values to the formula.

x		=	1\over{2}(x_1 +x_2)
x		=	1\over{2}(2 + 3)
x		=	1\over{2}(5)
x		=	2.5

y		=	1\over{2}(y_1 +y_2)
y		=	1\over{2}(4 + 0)
y		=	1\over{2}(4)
y		=	2

Thus, the coordinates are (4,0)

SLOPE OF A LINE

The slope (m)of the line is the rise over the unit of run.

Therefore, the formula in getting the slope is:

For example:

Find the slope of the line that passes through points P (3,2) and Q (-5,-4). To solve, first we assign the values according to the formula. So, let (x₁, y₁) be (3,2), (x₂, y₂) be (-5, -4). Then, substitute the following values into the formula.

Another example:

find the slope of the line that passes through points Q ( -5,3) and R ( 3, -2). Let, (x₁, y₁) be (-5, 3), (x₂, y₂) be (3, -2).

Slopes vary depending on to which the line rises.

* If a line rises to the right, then the slope is positive.

* If a line rises to the left, then the slope is negative

* If a line is horizontal then the slope is zero

* If a line is vertical its slope is undefined.

Try to get the slopes of the following lines:

1. S (5,5), T (1,1)

2. H (-4,5), I (3, -5)

3. K (1,5) L (1,-2)

4. O (-5, -4), (3, -4)