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A map of three public schools was created using a coordinate plane where the origin represents the center of the town. Euclid Elementary School is graphed at (−4, 1), Math Middle School is graphed at (1, 1), and Hypotenuse High School is graphed at (−4, −3). Each unit on the graph represents 1 mile.

Part A: Find the shortest distance, in miles, from Euclid Elementary School to Math Middle School. Show every step of your work. (2 points)

Part B: Find the shortest distance, in miles, from Euclid Elementary School to Hypotenuse High School. Show every step of your work. (2 points)

Part C: Find the shortest distance, in miles, from Math Middle School to Hypotenuse High School. Show every step of your work. (4 points)

1 answer

To find the shortest distances between the schools, we will use the distance formula, which is given by:

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

where \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points.

Part A: Find the shortest distance from Euclid Elementary School to Math Middle School

  1. Identify the coordinates:

    • Euclid Elementary School: \((-4, 1)\)
    • Math Middle School: \((1, 1)\)

  2. Plug into the distance formula: \[ d = \sqrt{(1 - (-4))^2 + (1 - 1)^2} \] Simplifying inside the parentheses: \[ d = \sqrt{(1 + 4)^2 + (1 - 1)^2} = \sqrt{(5)^2 + (0)^2} = \sqrt{25} \]

  3. Calculate the distance: \[ d = 5 \text{ miles} \]

Part B: Find the shortest distance from Euclid Elementary School to Hypotenuse High School

  1. Identify the coordinates:

    • Euclid Elementary School: \((-4, 1)\)
    • Hypotenuse High School: \((-4, -3)\)

  2. Plug into the distance formula: \[ d = \sqrt{(-4 - (-4))^2 + (-3 - 1)^2} \] Simplifying inside the parentheses: \[ d = \sqrt{(0)^2 + (-3 - 1)^2} = \sqrt{(0)^2 + (-4)^2} = \sqrt{16} \]

  3. Calculate the distance: \[ d = 4 \text{ miles} \]

Part C: Find the shortest distance from Math Middle School to Hypotenuse High School

  1. Identify the coordinates:

    • Math Middle School: \((1, 1)\)
    • Hypotenuse High School: \((-4, -3)\)

  2. Plug into the distance formula: \[ d = \sqrt{(-4 - 1)^2 + (-3 - 1)^2} \] Simplifying inside the parentheses: \[ d = \sqrt{(-5)^2 + (-4)^2} = \sqrt{25 + 16} = \sqrt{41} \]

  3. Calculate the distance: \[ d \approx 6.4 \text{ miles} \quad (\text{to the nearest tenth, } 6.4) \]

Summary of Results

  • Part A: The shortest distance from Euclid Elementary School to Math Middle School is 5 miles.
  • Part B: The shortest distance from Euclid Elementary School to Hypotenuse High School is 4 miles.
  • Part C: The shortest distance from Math Middle School to Hypotenuse High School is approximately 6.4 miles.