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: The shortest distance from Euclid Elementary to Math Middle is 8 miles. Part B: The shortest distance from Euclid Elementary to Hypotenuse High is 10 miles. Part C: The shortest distance from Math Middle to Hypotenuse High is approximately √164 miles. Part D: Zion traveled the shorter distance, about 12.81 miles.

Part A: Shortest Distance from Euclid Elementary to Math Middle

Using the distance formula (d = √((x₂ - x₁)² + (y₂ - y₁)²)):

The distance (d) from Euclid Elementary to Math Middle (EM) is √((6 - (-2))² + (6 - 6)²) = √(8²) = 8.

Part B: Shortest Distance from Euclid Elementary to Hypotenuse High

The distance (d) from Euclid Elementary to Hypotenuse High (EH) is √(((-2) - (-2))² + ((-4) - 6)²) = √(10²) = 10.

Part C: Shortest Distance from Math Middle to Hypotenuse High

The distance (d) from Math Middle to Hypotenuse High (MH) is √((6 - (-2))² + (6 - (-4))²) = √(8² + 10²) = √164.

Part D: Shay vs. Zion

Shay's path: EH + EM = 10 + 8 = 18 miles.

Zion's straight path from Hypotenuse High to Math Middle: dMH = √164 miles.

To determine who traveled the shortest distance, compare 18 to √164. As √164 ≈ 12.81, Zion traveled the shorter distance.