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

Posted on Aug 23, 2024 in Mathematics

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 y. A positive x moves right, a negative x moves left. Similarly, a positive y moves upward, and a negative y moves downward.

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 x-axis (negative value).
• Move 5 units down on the y-axis (negative value).

To locate point B (-3, 2):

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

To locate point C (4, 3):

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

To locate point D (2, -3):

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

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 (x₁, y₁) and Z (x₂, y₂), the distance formula is:

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) => x₁= -4, y₁ = -5
• Point 2: B(-3,2) => x₂= -3, y₂ = 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(x₁,y₁) and B(x₂,y₂), and a point C(x,y) that divides the segment in the ratio r, the coordinates of 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 .