Math Refresher

Coordinate Geometry
 Learning Objectives

By the end of this lesson, you will be able to:

Define the coordinate plane and the coordinates of a point in

coordinate geometry

Understand the formulas for distance and slope used in coordinate

geometry

List the various coordinate geometry formulas and their

components

Explain the fundamentals of coordinate geometry and its uses

Introduction to Coordinate Geometry
 Coordinate Geometry

Coordinate geometry involves studying geometric figures by plotting them on

coordinate axes.

It is a branch of mathematics that enables the

representation of geometric figures on a two-
dimensional plane and the understanding of their
properties.

Straight lines, curves, circles, ellipses, hyperbolas, and polygons can be accurately drawn and
represented to scale on the coordinate axes.
 Coordinate Geometry

Coordinate geometry also facilitates algebraic computations and the analysis of

geometric figure attributes using the coordinate system.

The starting points of the coordinate system are the zero degree
of Greenwich Longitude and the zero degree of Equator Latitude.
 Coordinate Plane

A Cartesian plane divides the plane space into two dimensions, making it convenient to
locate points.

The two axes of the coordinate plane are the horizontal

x-axis and the vertical y-axis.
 Coordinate Plane

Other features of the coordinate plane are as follows:

The two coordinate axes divide the plane into four

quadrants, and the point at which they intersect is
called the origin (0, 0).

In the coordinate plane, points are represented by

coordinate pairs (x, y), with x denoting the position along
the x-axis and y denoting the position along the y-axis.
 Coordinate Plane: Key Terms

Here are some key terms associated with a coordinate plane:

• Distance: Using the distance formula, derived from the

8 Pythagorean theorem, the distance between two points is
6
4
calculated on a graph.
(4,2)
2 • Slope: The slope of a line, denoted m, represents its
-6 -4 -2 2 4 6 8 steepness, determined by the ratio of vertical to horizontal
-X -2 X
-4
change.
-6 • Graphing: The coordinate plane visually depicts functions,
-8 equations, and shapes by plotting points and connecting
them with lines or curves.
 Coordinate Plane: Properties

The properties of points in the four quadrants of the coordinate plane are as follows:

8
6 • The origin O with coordinates (0, 0) represents the
Quadrant II Quadrant I
4 point of intersection between the x-axis and the y-
(-,+) (+,+)
2 axis.
-6 -4 -2 2 4 6 8 • The x-axis to the right of the origin O is the positive
-2
x-axis, while the x-axis to the left of the origin O is
-4
Quadrant III Quadrant IV the negative x-axis.
-6
(-,-) (+,-)
-8
 Coordinate Plane: Properties

The properties of points in the four quadrants of the coordinate plane are as follows:

8
6 • The y-axis above the origin O is the positive y-axis,
Quadrant II Quadrant I
4 while the y-axis below the origin O is the negative
(-,+) (+,+)
2 y-axis.
-6 -4 -2 2 4 6 8 • A point in the first quadrant (x, y) has positive
-2
values for both x and y; it is plotted with regards to
-4
Quadrant III Quadrant IV the positive x-axis and positive y-axis.
-6
(-,-) (+,-)
-8
 Coordinate Plane: Properties

The properties of points in the four quadrants of the coordinate plane are as follows:

• In the second quadrant, a point is represented as (-x,

8
y) and is plotted with reference to the negative x-axis
6
Quadrant II Quadrant I and positive y-axis.
4
(-,+) (+,+) • In the third quadrant, a point is represented as (-x, -y)
2
and is plotted with reference to the negative x-axis
-x -6 -4 -2 2 4 6 8 x
-2 and negative y-axis.

Quadrant III
-4
Quadrant IV • In the fourth quadrant, a point is represented as (x, -
-6 y) and is plotted with reference to the positive x-axis
(-,-) (+,-)
-8 and negative y-axis.
-y
 Coordinates of a Point

A coordinate is an address that helps locate a point in space.

8
X-coordinate
6
4
(4,2) • In a two-dimensional space, the coordinates of
2 Y-coordinate a point are denoted as (x, y).
-6 -4 -2 2 4 6 8 • Abscissa refers to the x value in a point (x, y)
-x -2 x
and represents the distance of the point along
-4
the x-axis from the origin.
-6
-8

-y
 Coordinates of a Point

A coordinate is an address that helps locate a point in space.

8 • Ordinate refers to the y value in a point (x,

X-coordinate
6 y) and represents the perpendicular
4 distance of the point from the x-axis,
(4,2)
2 Y-coordinate parallel to the y-axis.

-6 -4 -2 2 4 6 8 • The coordinates of a point are essential for

-x -2 x
performing various operations such as
-4
calculating distance, finding the midpoint,
-6
determining the slope of a line, and
-8
deriving the equation of a line.

-Y
Coordinate Geometry Formulas
 Euclidean Distance Formula

The distance formula calculates the distance between two points. It represents the length of the line
segment that connects them.

Hypotenuse
Y
(x2, y2)

(x1, y1) X
 Euclidean Distance Formula

Let's consider two points, A and B, with coordinates (x1, y1) and (x2, y2), respectively.

The distance between these two points is

calculated as follows:

D= x2 − x1 2 + y2 − y1 2

Where: 𝑥2 − 𝑥1 = Change in x
𝑦2 − 𝑦1 = Change in y
 Manhattan Distance Formula

In some cases, the distance is calculated using the Manhattan distance, also known as the taxicab
distance.

This distance is determined by summing the absolute differences in the x-

coordinates and y-coordinates between the two points.
 Manhattan Distance Formula

Consider two points, A and B, with coordinates (x1,y1) and (x2,y2), respectively.

d(A, B) = |x₂ - x₁| + |y₂ - y₁|

The distance between these
two points is calculated as
Where: 𝑥2 − 𝑥1 = Change in x
follows:
𝑦2 − 𝑦1 = Change in y
 Minkowski Distance Formula

The Minkowski distance is a generalized distance formula that combines

the Euclidean distance and Manhattan distance.
Y
(x2, y2)

(x1, y1) X

It is defined as the pth root of the sum of the absolute differences raised to the
power of p.
 Minkowski Distance Formula

Consider two points, A and B, with coordinates (x1, y1) and (x2, y2), respectively.

d(A, B) = ((|x₂ - x₁|)p + (|y₂ - y₁|)p)1/p

The distance between these two points
is calculated as follows: Where: 𝑥2 − 𝑥1 = Change in x
𝑦2 − 𝑦1 = Change in y
 Midpoint Formula

The midpoint formula is used to find the coordinates of the midpoint

between two points in a coordinate plane.

Y
(x2, y2)

(x1, y1) X
 Midpoint Formula

Consider two points, A and B, with coordinates (x1, y1) and

(x2, y2), respectively.

The coordinates of the

midpoint, denoted as M(x, y), (x1+x2) (y1+y2)
can be calculated using the
M x, y = ( , )
2 2
following formula:
Coordinate Geometry: Line
 Line

In coordinate geometry, a line is a straight path that extends infinitely

in both directions.

(x2, y2)

(x1, y1)
 Line

The equation of a line relates the x and y coordinates of points on the line.

(x2, y2)

The y-intercept of a line is the point at which

(x1, y1)
it intersects the y-axis.

The slope of a line represents its steepness or inclination and is defined as the ratio of
the vertical change (rise) to the horizontal change (run) between two points on the line.
 Equation of a Line

The table below provides the formulae for different forms of a line:

Formula Name Formula

General Formula Ax +By = C

Slope Intercept Form Y = mx + b

Point Slope Form (y – y1) = m(x – x1)

Here, m is the slope and b is the y-intercept.

 Slope Formula

The slope of a line, often referred to as its gradient, is a numerical

representation of the line's steepness and direction.

x2, y2
• It is calculated as the difference between the
change in the x and y coordinates.
y2 – y1 • The slope can be determined by selecting any
θ two consecutive points on the line or using the
x1, y1
angle that the line forms with the positive x-axis.
x2 – x1

The slope of the vertical line at an angle θ with the x-axis vertical is, m = tan(θ).
 Slope Formula

For a line segment formed by two points, the slope at an angle θ can be calculated using the following
formula:

x2, y2

y2 −y1
tan(θ) =
y2 – y1 x2 −x1
θ
x1, y1
m = tan(θ)
x2 – x1

θ
 Distance of a Point from Line

Consider a point P(x₀, y₀) and a line represented by the equation Ax + By + C = 0.

x0, y0
The distance between the point and the line can
be calculated using the following formula:
θ

|Ax0 + By0 + C|
a2 + b2
 Applications of Measuring the Distance

Geometric analysis Line fitting

To determine the spatial relationship To evaluate the accuracy of regression

and proximity between a point and a models by calculating the distance
line between data points and the fitted line

Error estimation Optimization problems

To measure accuracy or deviation in To minimize or maximize criteria

numerical approximations or by calculating distances to
computational algorithms constraint lines
 Key Takeaways

Coordinate geometry is a branch of mathematics that examines

geometric figures using a coordinate system and aids in algebraic
computations.

The distance formula is used to determine the distance between

two points on a coordinate plane.

The coordinate plane divides the plane space into two dimensions
via the x and y axes, which further divide the plane into four
quadrants. The point of intersection of these axes is known as the
origin (0,0).
 Key Takeaways

The slope of a line provides a numerical representation of both its

steepness and direction.

The midpoint formula is employed to determine a point that

lies exactly halfway between two points on a coordinate
plane.
Knowledge Check
Knowledge
Check
The distance of the point P(2, 3) from the x-axis is
1

A. 2

B. 3

C. 1

D. 5
 Knowledge
Check
The distance of the point P(2, 3) from the x-axis is
1

A. 2

B. 3

C. 1

D. 5

The correct answer is B

The distance of a point from the x-axis is equal to its ordinate, which is 3.
Knowledge
Check
The distance of the point (α, β) from the origin is
2

A. ∝ +β

B. ∝ 2 + β2

C. | ∝ | + |β|

D. ∝ 2 + β2
 Knowledge
Check
The distance of the point (α, β) from the origin is
2

A. ∝ +β

B. ∝ 2 + β2

C. | ∝ | + |β|

D. ∝ 2 + β2

The correct answer is D

The distance of the point (α, β) from the origin can be calculated using the formula ∝ −0 2 + 𝛽−0 2

= ∝ 2+ 𝛽 2
Knowledge
Check
What is the distance between the point P(4, -2) and the line 2x + 3y - 6 = 0?
3

2
A.
13
3
B.
13

4
C.
13

6
D.
13
 Knowledge
Check
What is the distance between the point P(4, -2) and the line 2x + 3y - 6 = 0?
3

2
A.
13
3
B.
13

4
C.
13

6
D.
13

The correct answer is C

The distance can be calculated using the formula:|Ax₁ + By₁ + C| / √(A² + B²). In this case, A = 2, B = 3, and C = -6.
Distance = |2(4) + 3(-2) - 6| / √(2² + 3²) = |8 - 6 - 6| / √(4 + 9) = |-4| / √13 = 4 / √13.
Knowledge
Check
Which of the following points has a slope of 2 with respect to the origin (0, 0)?
4

A. (2, 6)

B. (-3, 6)

C. (-1, -2)

D. (0, 3)
 Knowledge • 4
Check
Which of the following points has a slope of 2 with respect to the origin (0, 0)?
4

A. (2, 6)

B. (-3, 6)

C. (-1, -2)

D. (0, 3)

The correct answer is C

The slope is calculated using the formula (y-0)/(x-0) = (-2 - 0) / (-1 - 0) = -2 / -1 = 2. Hence, the answer is C.
Knowledge
Check
What is the midpoint between the points A (2, 5) and B (8, -1)?
5

A. (5, 2)

B. (6, 3)

C. (5, -3)

D. (4, 1)
 Knowledge • 5
Check
What is the midpoint between the points A (2, 5) and B (8, -1)?
5

A. (5, 2)

B. (6, 3)

C. (5, -3)

D. (4, 1)

The correct answer is A

Midpoint is calculated using the formula [(x1+x2)/2, (y1+y2)/2] = ((2 + 8) / 2, (5 + (-1)) / 2) = (10 / 2, 4 / 2) = (5, 2)