Spatial reasoning – cardinal directions among three towns Given: Middletown is north of Centerville. Centerville is east of Penfield. Question: If those two facts are true, is the statement “Penfield is northwest of Middletown” true, false, or uncertain?

Introduction / Context:
This problem checks your 2D spatial reasoning with compass directions. By assigning simple coordinates, you can test relative positions rigorously instead of relying on mental sketches that can be error-prone.

Given Data / Assumptions:

• Middletown is north of Centerville (same east–west line, greater northing).
• Centerville is east of Penfield (same north–south line, greater easting).
• Claim to evaluate: “Penfield is northwest of Middletown.”

Concept / Approach:
Assign coordinates: let Centerville be at (c, y0). Then Middletown is at (c, y0 + d) where d > 0. Since Centerville is east of Penfield, place Penfield at (p, y0) with p < c. Compare Penfield to Middletown: vector from Middletown to Penfield is (p - c, y0 - (y0 + d)) = (negative, negative) ⇒ southwest, not northwest.

Step-by-Step Solution:
1) Place Centerville C = (0, 0) without loss of generality.2) Middletown M is north of C ⇒ M = (0, +1).3) Penfield P is west of C ⇒ P = (-1, 0).4) Compare P relative to M: Δx = -1 - 0 = -1 (west), Δy = 0 - 1 = -1 (south). So P is southwest of M.5) Therefore the statement “Penfield is northwest of Middletown” is False.

Verification / Alternative check:
Any consistent distances keep the signs the same: west of C enforces x lower than C; north of C enforces y higher than C. Hence relative to M, P is always southwest.

Why Other Options Are Wrong:

• True: Contradicts the coordinate reasoning.
• Uncertain: Directions are fully determined by the given relations.
• Both true and false: Not a valid single outcome.

Common Pitfalls:
Accidentally swapping east–west or north–south, or placing towns on diagonals that ignore the “same line” implication in directional statements.

Final Answer:
False