The straight-line distance between Town A and Town B is 15 km. Most travelers passing through the towns, however, use...

Phase 1: Owning the Dataset

Visualization Selection

This is a geometry problem involving distances and paths, so let's draw a diagram showing the towns and highways.

Visual Representation

        Town B
         |
         | y km (north)
         |
Town A---●---Intersection
    x km (west)
    
    Straight-line: 15 km

The travelers go from Town A west to an intersection, then north to Town B. The straight-line distance from A to B is 15 km.

Phase 2: Understanding the Question

Key Relationships

1. Right Triangle Formation: The path forms a right triangle where:
   • One leg = x km (horizontal distance west)
   • Other leg = y km (vertical distance north)
   • Hypotenuse = 15 km (straight-line distance)
2. Pythagorean Theorem: \(\mathrm{x}^2 + \mathrm{y}^2 = 15^2 = 225\)
3. Distance Constraint: "The intersection is farther from Town A than Town B"
   • Distance from A to intersection = x
   • Distance from B to intersection = y
   • Therefore: \(\mathrm{x} > \mathrm{y}\)

What We're Looking For

We need values of x and y from the choices [5, 8, 9, 10, 12] that:

• Satisfy \(\mathrm{x}^2 + \mathrm{y}^2 = 225\)
• Meet the condition \(\mathrm{x} > \mathrm{y}\)

Phase 3: Finding the Answer

Systematic Check

Let's check which combinations work:

If x = 12:

• \(12^2 + \mathrm{y}^2 = 225\)
• \(144 + \mathrm{y}^2 = 225\)
• \(\mathrm{y}^2 = 81\)
• \(\mathrm{y} = 9\) ✓

Does this satisfy \(\mathrm{x} > \mathrm{y}\)? Yes: \(12 > 9\) ✓

Verification (use calculator if needed):

• \(12^2 + 9^2 = 144 + 81 = 225\) ✓
• This equals \(15^2\) ✓

? Stop here - we found our answer.

Phase 4: Solution

Final Answer

• x = 12 (distance west from Town A to intersection)
• y = 9 (distance north from intersection to Town B)

These values satisfy both requirements:

1. They form a right triangle with hypotenuse 15 km (\(12^2 + 9^2 = 225 = 15^2\))
2. The intersection is farther from Town A than Town B (\(12 > 9\))