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 (−2, 6), Math Middle School is graphed at (6, 6), and Hypotenuse High School is graphed at (−2, −4). 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)

Part D: Shay traveled from Hypotenuse High to Euclid Elementary and then to Math Middle. Zion traveled from Hypotenuse High to Math Middle along a straight path. Who went the shortest distance? Explain. (4 points)

Answer 1

Part A: Euclid Elementary School to Math Middle School is 8 miles

Part B: Euclid Elementary School to Hypotenuse High School is 10 miles

Part C: Math Middle School to Hypotenuse High School is 12.81 miles

Part D: Zion traveled the shortest distance because
`\(12.81 \text{ miles} < 18 \text{ miles}\).`

How did we get the value?

To find the distances between the schools, use the distance formula:

`\[ \text{Distance} = √( (x_2 - x_1)^2 + (y_2 - y_1)^2 ) \]`

Calculate the distances:

Part A: Euclid Elementary School to Math Middle School

`\[ \text{Distance} = √( (6 - (-2))^2 + (6 - 6)^2 ) \]`

`\[ \text{Distance} = √( 8^2 + 0^2 ) \]`

`\[ \text{Distance} = √( 64 ) \]`

`\[ \text{Distance} = 8 \text{ miles} \]`

Part B: Euclid Elementary School to Hypotenuse High School

`\[ \text{Distance} = √( ((-2) - (-2))^2 + ((-4) - 6)^2 ) \]`

`\[ \text{Distance} = √( 0^2 + (-10)^2 ) \]`

`\[ \text{Distance} = √( 100 ) \]`

`\[ \text{Distance} = 10 \text{ miles} \]`

Part C: Math Middle School to Hypotenuse High School

`\[ \text{Distance} = √( ((-2) - 6)^2 + ((-4) - 6)^2 ) \]`

`\[ \text{Distance} = √( (-8)^2 + (-10)^2 ) \]`

`\[ \text{Distance} = √( 64 + 100 ) \]`

`\[ \text{Distance} = √( 164 ) \]`

This distance is approximately
`\(12.81\)` miles.

Part D: Shay vs. Zion

Shay traveled from Hypotenuse High to Euclid Elementary and then to Math Middle:

`\[ \text{Total distance for Shay} = 10 + 8 = 18 \text{ miles} \]`

Zion traveled from Hypotenuse High to Math Middle along a straight path:

`\[ \text{Distance for Zion} = √( (6 - (-2))^2 + ((-4) - 6)^2 ) \]`

`\[ \text{Distance for Zion} = √( 8^2 + (-10)^2 ) \]`

`\[ \text{Distance for Zion} = √( 164 ) \]`

This distance is approximately
`\(12.81\)` miles.

Therefore, Zion traveled the shortest distance because
`\(12.81 \text{ miles} < 18 \text{ miles}\).`