1. **State the problem:** We need to find the distance between two points, X and Y, in Manhattan, where streets run east-west and avenues run north-south. Point X is at the junction of 15th Street and 7th Avenue. Point Y is at the junction of 21st Street and 4th Avenue. The distance between streets (north-south direction) is 80 m. The distance between avenues (east-west direction) is 260 m. We aim to find: a) The straight-line (Euclidean) distance between X and Y. b) The shortest distance traveling along the streets and avenues (Manhattan distance). 2. **Determine the coordinate differences:** - Streets run east-west; numbering increases from south to north. - Avenues run north-south; numbering increases from east to west. - For point X at 15th Street and 7th Avenue: - Street coordinate: 15 - Avenue coordinate: 7 - For point Y at 21st Street and 4th Avenue: - Street coordinate: 21 - Avenue coordinate: 4 3. **Calculate the differences in street and avenue counts:** - Difference in streets: $21 - 15 = 6$ - Difference in avenues: $7 - 4 = 3$ 4. **Convert differences to meters:** - Vertical distance (north-south): $6 \times 80 = 480$ m - Horizontal distance (east-west): $3 \times 260 = 780$ m 5. **Calculate the straight-line distance (a):** Use Pythagoras' theorem: $$ d = \sqrt{(480)^2 + (780)^2} = \sqrt{230400 + 608400} = \sqrt{838800} $$ $$ d \approx 915.9 \text{ m} $$ Rounded to the nearest metre: **$916$ m** 6. **Calculate the shortest distance along the roads (b):** This is the Manhattan distance: $$ d_{Manhattan} = 480 + 780 = 1260 \text{ m} $$ **Final answers:** a) Straight-line distance is **916 m**. b) Shortest road distance is **1260 m**.