Question

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)

Answer 1

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**.