Problem Summary:

You are asked to calculate the distance between two points, X and Y, using both the straight-line distance (part a) and the shortest distance along the roads (part b).

• Point X: At the junction of 18th Street and 7th Avenue.
• Point Y: At the junction of 24th Street and 2nd Avenue.

Road Grid Information:

• Distance between streets (north-south): 80 m
• Distance between avenues (east-west): 260 m

Part (a): Straight-Line Distance

The straight-line distance can be calculated using the Pythagorean theorem. We need to calculate the horizontal and vertical differences between the points, treat them as the two legs of a right triangle, and solve for the hypotenuse.

• Horizontal distance: From 7th Avenue to 2nd Avenue $\text{Distance} = (7 - 2) \times 260\,\text{m} = 5 \times 260\,\text{m} = 1300\,\text{m}$
• Vertical distance: From 18th Street to 24th Street $\text{Distance} = (24 - 18) \times 80\,\text{m} = 6 \times 80\,\text{m} = 480\,\text{m}$

Now, using the Pythagorean theorem: $\text{Straight-line distance} = \sqrt{(1300)^2 + (480)^2}$

Part (b): Shortest Distance Along the Roads

For the shortest road distance, we add the horizontal and vertical distances directly (since you're constrained to follow the roads):

$\text{Shortest road distance} = 1300\,\text{m} + 480\,\text{m} = 1780\,\text{m}$

I'll now calculate the straight-line distance for you.### Solutions:

• (a) Straight-line distance: The straight-line distance between Point X and Point Y is approximately 1386 meters.
• (b) Shortest road distance: The shortest distance along the roads is 1780 meters.

Would you like further clarification on any part of the solution?

Follow-up questions:

1. How would the distances change if the grid spacing between streets was different?
2. Can you apply the same method for diagonal streets or avenues?
3. How do these calculations relate to real-world city block layouts?
4. Can you think of other real-life applications of the Pythagorean theorem?
5. What would happen if the distance between avenues was variable?

Tip: For grid-like layouts in city maps, the shortest path along streets is often called the "Manhattan distance."