Scenario: Navigation & Mapping - Learn coordinate geometry through GPS navigation and map reading!
You're using a GPS navigation system to plan a trip. You need to find the distance between two cities and determine the midpoint for a rest stop. The cities are located at coordinates A(3, 4) and B(7, 8).
Navigation Challenge:
Find the distance between cities A(3, 4) and B(7, 8), and locate the midpoint for a rest stop.
Coordinate plane showing points A and B
Coordinate Format: (x, y) where x is horizontal position and y is vertical position
(+, +) - Top right
Example: (3, 4)
(-, +) - Top left
Example: (-3, 4)
(-, -) - Bottom left
Example: (-3, -4)
(+, -) - Bottom right
Example: (3, -4)
Distance Formula: d = √[(x₂ - x₁)² + (y₂ - y₁)²]
This formula comes from the Pythagorean theorem applied to the coordinate plane.
Given: A(3, 4) and B(7, 8)
Step 1: Find differences
x₂ - x₁ = 7 - 3 = 4
y₂ - y₁ = 8 - 4 = 4
Step 2: Apply distance formula
d = √(4² + 4²) = √(16 + 16) = √32 ≈ 5.66 units
Midpoint Formula: M = ((x₁ + x₂)/2, (y₁ + y₂)/2)
The midpoint is exactly halfway between two points.
Given: A(3, 4) and B(7, 8)
Step 1: Calculate x-coordinate
x = (3 + 7)/2 = 10/2 = 5
Step 2: Calculate y-coordinate
y = (4 + 8)/2 = 12/2 = 6
Answer: Rest stop should be at M(5, 6)
Find the distance between points P(1, 2) and Q(5, 6).
Your solution:
Find the midpoint between R(-2, 3) and S(4, -1).
Your solution:
A delivery truck needs to go from warehouse A(0, 0) to store B(8, 6). Find the distance and midpoint for a fuel stop.
Your calculation: