Identify the types of shortest paths between two points on and through the Earth's surface | Step-by-Step Solution
Problem
What is the name of the 'total' shortest path between two points, regardless of adherence to surfaces? If the green arc between Sydney, Australia, and Lima, Peru, is a geodesic, since it follows Earth's surface, what is the red line called (the shortest path possible period since it would be a tunnel through the earth)?
🎯 What You'll Learn
- Understand different types of shortest paths on a sphere
- Learn about geodesic and chord lines
- Explore geometric principles of global navigation
Prerequisites: Basic understanding of spherical geometry, Concept of shortest distance
💡 Quick Summary
Hi there! This is a great geometry and geography question about different ways we can think about "shortest distance" between two points on Earth. When you're looking at these two different paths, think about what constraints or limitations each one has - what if you had to stay on the Earth's surface versus what if you could travel in a completely straight line with no obstacles? You've already correctly identified that the curved path along the surface is called a geodesic, so now consider what we call a straight line that connects two points on a circle or sphere in geometry class. What term do you remember for the straight line segment that "cuts across" the interior of a circle, connecting two points on the circumference? Try thinking back to your geometry lessons about circles and the different line segments we can draw - there's a specific name for this type of straight-through connection!
Step-by-Step Explanation
Understanding Different Types of Shortest Paths
What We're Solving:
You're asking about two different types of shortest paths between two points on Earth - one that follows the curved surface (the green arc/geodesic) and one that goes straight through the planet (the red line). We need to identify what each path is called!The Approach:
This is about understanding different ways to measure "shortest distance" depending on the constraints we have. If you're walking, you follow the surface, but if you could magically tunnel through, you'd go straight. Each has its own mathematical name!Step-by-Step Solution:
Step 1: Understanding the surface path (green arc)
- You correctly identified this as a geodesic
- A geodesic is the shortest distance between two points when you must stay on a curved surface
- On Earth's sphere, this creates what looks like an arc when viewed from space
- It's like stretching a rubber band over a globe between two points
- This is the shortest possible distance if there are no constraints at all
- It goes directly through the Earth in a straight line
- This represents the true geometric shortest distance in 3D space
- The straight-line path through the Earth is called a chord
- In geometry, a chord is any straight line that connects two points on a circle (or sphere)
- Some sources might also call this the "Euclidean distance" since it follows straight-line geometry
The Answer:
The red line (straight path through Earth) is called a chord. While the green arc is a geodesic (shortest path along the surface), the chord represents the absolute shortest distance between the two points in three-dimensional space.Memory Tip:
Think of it like a bow and arrow: the geodesic is like the curved bow, while the chord is like the straight bowstring connecting the two ends. The chord always "cuts through" to make the shortest possible path!Great question - understanding the difference between surface distances and straight-line distances is fundamental in both geography and geometry! 🌍
⚠️ Common Mistakes to Avoid
- Assuming the shortest path is always a straight line on a flat map
- Misunderstanding the difference between surface and direct paths
- Confusing geodesic routes with map projections
This explanation was generated by AI. While we work hard to be accurate, mistakes can happen! Always double-check important answers with your teacher or textbook.
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📷 Problem detected:
Solve: 2x + 5 = 13
Step 1:
Subtract 5 from both sides...
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