Combine the power of algebra and geometry! Learn to solve complex problems by integrating algebraic equations with geometric principles and coordinate systems.
Problem: A rectangle has vertices at A(2,3), B(8,3), C(8,7), and D(2,7). Find the area and perimeter of the rectangle.
Step 1: Plot the points and identify the rectangle
A(2,3), B(8,3), C(8,7), D(2,7)
Length = 8-2 = 6 units, Width = 7-3 = 4 units
Step 2: Calculate area and perimeter
Area = length × width = 6 × 4 = 24 square units
Perimeter = 2(length + width) = 2(6 + 4) = 20 units
Problem: Find the distance between points P(1,2) and Q(5,6), and the midpoint of segment PQ.
Step 1: Use distance formula
d = √[(5-1)² + (6-2)²] = √[4² + 4²] = √32 = 4√2 ≈ 5.66
Step 2: Find midpoint
Midpoint = ((1+5)/2, (2+6)/2) = (3, 4)
Problem: Find the point of intersection of lines y = 2x + 1 and y = -x + 4.
Step 1: Set equations equal to each other
2x + 1 = -x + 4
Step 2: Solve for x
2x + x = 4 - 1
3x = 3
x = 1
Step 3: Find y-coordinate
y = 2(1) + 1 = 3
Intersection point: (1, 3)
Scenario: A ship starts at point A(0,0) and travels to point B(12,5). If the ship can travel at 13 km/h, how long will the journey take?
Step 1: Calculate distance using distance formula
d = √[(12-0)² + (5-0)²] = √[144 + 25] = √169 = 13 km
Step 2: Calculate time
Time = Distance ÷ Speed = 13 km ÷ 13 km/h = 1 hour
Definition:
x = f(t), y = g(t)
Use: Describing curves and motion in coordinate systems
Components:
v = (x, y) = xi + yj
Use: Force analysis and motion in physics
Structural Design
Load distribution and stress analysis
Bridge construction and architectural planning
3D Modeling
Coordinate transformations and rendering
Game development and animation systems
Path Planning
Optimal route calculation
GPS systems and autonomous vehicle control