# Understanding the Cartesian Plane: Distance, Area, and Point Division

## 1.1 Location of Given Points in the Cartesian Plane

A plane, ** xy**, formed by two perpendicular lines, is divided into four quadrants (I, II, III, IV).

The intersection point of these lines is called the origin (** O**), with coordinates (

**0, 0**).

From the origin, you can observe positive and negative directions. Any point **P** in the plane is represented by ordered pairs *( x, y)*. This system, used in analytic geometry, is known as the

**Cartesian plane**.

To locate a point, distinguish the values of ** x** and

**. A positive**

*y***moves right, a negative**

*x***moves left. Similarly, a positive**

*x***moves upward, and a negative**

*y***moves downward.**

*y*### Example

Locate the points: ** A(-4,-5), B(-3,2), C(4,3), D(2,-3)**.

#### Steps

#### Procedure

To locate point A (**-4, -5**):

- Start at origin
.*O* - Move
**4**units left on the-axis (negative value).*x* - Move
**5**units down on the-axis (negative value).*y*

To locate point B (**-3, 2**):

- Start at origin
.*O* - Move
**3**units left on the-axis (negative value).*x* - Move
**2**units up on the-axis (positive value).*y*

To locate point C (**4, 3**):

- Start at origin
.*O* - Move
**4**units right on the-axis (positive value).*x* - Move
**3**units up on the-axis (positive value).*y*

To locate point D (**2, -3**):

- Start at origin
.*O* - Move
**2**units right on the-axis (positive value).*x* - Move
**3**units down on the-axis (negative value).*y*

#### Answer

## 1.2 Distance Between Two Points

The distance between two points is the length between them, always a positive value (absolute value).

Given points *W** ( x_{1}, y_{1})* and

*Z**(*, the distance formula is:

**x**)_{2}, y_{2}This formula derives from the **Pythagorean Theorem**, as illustrated below:

### Example

Calculate the distance between ** A (-4,-5)** and

**.**

*B(-3,2)*#### Steps

#### Procedure

Use the distance formula:

Assign points (order doesn’t affect the result):

- Point 1: A(-4,-5) => x
_{1}= -4, y_{1}= -5 - Point 2: B(-3,2) => x
_{2}= -3, y_{2}= 2

Substitute the values:

Apply the rules of signs and simplify:

#### Answer

The distance between the points is .

## 1.3 The Area of a Triangle Given Its Vertices

### Example

Calculate the area of the triangle with vertices: **A(0,0), B(3,2), and C(3,5)**.

#### Steps

#### Procedure

Sketch the triangle and introduce point **A’** to form a right triangle:

Calculate the area using geometric concepts:

**ABC** = Area of **AA’C** – Area of **AA’B**

Area of a triangle = (base * height) / 2

Area of **AA’C** (base = AA’, height = A’C):

Area of **AA’B** (base = AA’, height = A’B):

Area of **ABC** = Area of **AA’C** – Area of **AA’B**

#### Answer

The area of the triangle is .

## 1.4 Coordinates of the Point that Divides a Linear Segment in a Given Ratio

### Theorem of the Division of a Segment in a Given Ratio

Given a segment with endpoints ** A(x_{1},y_{1})** and

**, and a point**

*B(x*_{2},y_{2})**C(x,y)**that divides the segment in the ratio

**, the coordinates of**

*r***C**are:

### Point of Division of a Linear Segment

If **C** divides the segment in the ratio , its coordinates are:

### Midpoint of a Segment

When the ratio is 1:1 (), the formula simplifies to the midpoint formula:

### Example

Find the coordinates of point **A** that divides the segment **E(0, 4)** and **F(3,-3)** in the ratio .

#### Steps

#### Procedure

Calculate the ** x** coordinate:

Calculate the ** y** coordinate:

#### Answer

The point dividing **EF** in the ratio 3/4 is .