To find the shortest distances between the schools, we can use the distance formula. The distance \( d \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) in a coordinate plane is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Part A: Shortest distance from Euclid Elementary School to Math Middle School
Coordinates:
- Euclid Elementary School: \( (-2, 6) \)
- Math Middle School: \( (6, 6) \)
Distance Calculation:
\[ d = \sqrt{(6 - (-2))^2 + (6 - 6)^2} \] \[ d = \sqrt{(6 + 2)^2 + (0)^2} \] \[ d = \sqrt{(8)^2} \] \[ d = \sqrt{64} \] \[ d = 8 \]
Answer for Part A: The shortest distance from Euclid Elementary School to Math Middle School is 8 miles.
Part B: Shortest distance from Euclid Elementary School to Hypotenuse High School
Coordinates:
- Euclid Elementary School: \( (-2, 6) \)
- Hypotenuse High School: \( (-2, -4) \)
Distance Calculation:
\[ d = \sqrt{(-2 - (-2))^2 + (-4 - 6)^2} \] \[ d = \sqrt{(0)^2 + (-4 - 6)^2} \] \[ d = \sqrt{0 + (-10)^2} \] \[ d = \sqrt{100} \] \[ d = 10 \]
Answer for Part B: The shortest distance from Euclid Elementary School to Hypotenuse High School is 10 miles.
Part C: Shortest distance from Math Middle School to Hypotenuse High School
Coordinates:
- Math Middle School: \( (6, 6) \)
- Hypotenuse High School: \( (-2, -4) \)
Distance Calculation:
\[ d = \sqrt{(-2 - 6)^2 + (-4 - 6)^2} \] \[ d = \sqrt{(-8)^2 + (-10)^2} \] \[ d = \sqrt{64 + 100} \] \[ d = \sqrt{164} \] \[ d \approx 12.81 \]
Answer for Part C: The shortest distance from Math Middle School to Hypotenuse High School is approximately 12.81 miles.
Part D: Compare distances traveled by Shay and Zion
Shay's Route:
- Distance from Hypotenuse High to Euclid Elementary: 10 miles (from Part B).
- Distance from Euclid Elementary to Math Middle School: 8 miles (from Part A).
Total distance for Shay: \[ d_{Shay} = 10 + 8 = 18 \text{ miles} \]
Zion's Route:
- Direct distance from Hypotenuse High to Math Middle:
- We calculated this in Part C to be approximately \(12.81 \text{ miles}\).
Comparison:
- Shay traveled 18 miles.
- Zion traveled approximately 12.81 miles.
Conclusion: Zion went the shortest distance because approximately 12.81 miles is less than 18 miles.