Distances in a City A city has streets that run north and...

Distances in a City A city has streets that run north and south and avenues that run east and west, all equally spaced. Streets and avenues are numbered sequentially, as shown in the figure. The walking distance between points $A$ and $B$ is 7 blocks-that is, 3 blocks east and 4 blocks north. To find the straight-line distance $d,$ we must use the Distance Formula. (a) Find the straight-line distance (in blocks) between $A$ and $B$. (b) Find the walking distance and the straight-line distance between the comer of 4 th St. and 2 nd Ave. and the corner between the comer of 4 th Ave. (c) What must be true about the points $P$ and $Q$ if the walking distance between $P$ and $Q$ equals the straight-line distance between $P$ and $Q ?$

Distances in a City A city has streets that run north and south and avenues that run east and west, all equally spaced. Streets and avenues are numbered sequentially, as shown in the figure. The walking distance between points $A$ and $B$ is 7 blocks-that is, 3 blocks east and 4 blocks north. To find the straight-line distance $d,$ we must use the Distance Formula.
(a) Find the straight-line distance (in blocks) between $A$ and $B$.
(b) Find the walking distance and the straight-line distance between the comer of 4 th St. and 2 nd Ave. and the corner between the comer of 4 th Ave.
(c) What must be true about the points $P$ and $Q$ if the walking distance between $P$ and $Q$ equals the straight-line distance between $P$ and $Q ?$

Key Concepts

Equality Condition for Distances

This concept refers to the specific scenario when the Manhattan distance and Euclidean distance are equal. This occurs if and only if the movement between the two points is along a straight line in a single direction (either purely horizontal or purely vertical), meaning no right angle is introduced.

Pythagorean Theorem

A fundamental principle in geometry that relates the lengths of the sides of a right triangle. It is used in calculating the Euclidean distance by treating the differences in horizontal and vertical positions as the legs of a right triangle.

Grid Coordinate System

In many urban planning problems, streets and avenues are laid out in a grid, which simplifies distance calculations by aligning with the axes. This system allows for easy application of both Manhattan and Euclidean distance concepts.

Manhattan Distance

Also known as the grid or walking distance, Manhattan distance is the sum of the absolute differences of the coordinates. It represents how far apart two points are when movement is restricted to fixed horizontal and vertical paths, like city blocks.

Euclidean Distance

This is the straight-line distance between two points, calculated using the Distance Formula derived from the Pythagorean Theorem. It gives the shortest distance between the points, irrespective of any constraints that might force movement along specific paths.

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