5.1 Mapping the Plane
Imagine a giant grid covering your entire classroom floor. This is the coordinate plane - a magical map where we can locate any point using numbers!
1
The Horizontal Line (X-axis): Like the horizon where the sky meets the earth.
2
The Vertical Line (Y-axis): Stands tall like a tree growing upwards.
3
Origin (0,0): Where the two axes meet - the center of our mathematical universe!
The coordinate system is called "Cartesian" after René Descartes, who supposedly got the idea while watching a fly on his ceiling!
5.2 Devising a Coordinate System
Let's create our own coordinate system to locate any point precisely.
Quadrants
Coordinates
Practice
Quadrant I (+,+)
Quadrant II (-,+)
Quadrant III (-,-)
Quadrant IV (+,-)
1
The plane is divided into 4 quadrants by the X and Y axes.
2
Quadrants are numbered counter-clockwise starting from the top-right.
3
The signs of coordinates in each quadrant:
- Quadrant I: (+, +)
- Quadrant II: (-, +)
- Quadrant III: (-, -)
- Quadrant IV: (+, -)
1
Any point is written as (x, y) - called an ordered pair.
2
x-coordinate (abscissa): How far left or right from the origin.
3
y-coordinate (ordinate): How far up or down from the origin.
Point P = (x, y)
Example: The point (3, -2) means:
- 3 units to the right on X-axis
- 2 units down on Y-axis
- Located in Quadrant IV
Let's practice identifying coordinates:
From left to right:
- (-1, 2.5)
- (1, -2)
- (-2, 1)
5.3 Distance Between Two Points
How do we calculate how far apart two points are? Let's find out!
Distance Formula: √[(x₂ - x₁)² + (y₂ - y₁)²]
1
Imagine a right triangle formed by the two points.
2
The distance is the hypotenuse of this triangle!
Distance between A(2,3) and B(5,7) is 5 units!
This formula comes directly from the Pythagorean theorem (a² + b² = c²)!
5.4 Mid-point of a Line Segment
The exact middle point between two locations - like the perfect spot to share a pizza!
Midpoint Formula: [(x₁ + x₂)/2, (y₁ + y₂)/2]
1
Average the x-coordinates of the two points.
2
Average the y-coordinates of the two points.
Midpoint M of A(1,1) and B(5,5) is (3,3)
Example: For points A(1,1) and B(5,5):
- x-coordinate: (1 + 5)/2 = 3
- y-coordinate: (1 + 5)/2 = 3
- Midpoint M = (3, 3)
5.5 Points of Trisection
Dividing a line into three equal parts - like cutting a candy bar for three friends!
A(1,3)
B(7,9)
P(3,5)
Q(5,7)
Trisection Formulas:
First point: [(2x₁ + x₂)/3, (2y₁ + y₂)/3]
Second point: [(x₁ + 2x₂)/3, (y₁ + 2y₂)/3]
1
The first point divides the line in ratio 1:2
2
The second point divides the line in ratio 2:1
Trisection points of A(
1,3) and B(
7,9):
- First point P: (3,5)
- Second point Q: (5,7)
5.6 Section Formula
A general formula to divide a line in any ratio - the ultimate sharing tool!
Section Formula (dividing in ratio m:n):
P = [(mx₂ + nx₁)/(m+n), (my₂ + ny₁)/(m+n)]
1
The midpoint is just a special case where m:n = 1:1
2
Trisection points are cases of 1:2 and 2:1 ratios
Point dividing A(2,2) and B(8,8) in ratio 2:1 is (6,6)
Example: Find point P that divides A(2,2) and B(8,8) in ratio 2:1
- P_x = (2×8 + 1×2)/(2+1) = (16+2)/3 = 6
- P_y = (2×8 + 1×2)/(2+1) = (16+2)/3 = 6
- P = (6,6)
5.7 Coordinates of the Centroid
The balancing point of a triangle - where it would perfectly balance on a pencil tip!
A(1,1)
B(5,1)
C(3,5)
G(3,2.33)
Centroid Formula (of triangle ABC):
G = [(x₁ + x₂ + x₃)/3, (y₁ + y₂ + y₃)/3]
1
Average all three x-coordinates
2
Average all three y-coordinates
Centroid of triangle with vertices A(1,1), B(5,1), C(3,5) is (3,2.33)
The centroid is always located at the intersection point of the medians (lines from vertices to midpoints of opposite sides)!