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)
Part D: Deon traveled from Hypotenuse High to Euclid Elementary and then to Math Middle. Kaya traveled from Hypotenuse High to Math Middle along a straight path. Who went the shortest distance? Explain. (4 points)
Part A:
To find the shortest distance from Euclid Elementary School to Math Middle School, we can use the distance formula. The distance formula is given by:
d = √((x2 - x1)^2 + (y2 - y1)^2)
Using the coordinates of Euclid Elementary School as (-4, 1) and Math Middle School as (1, 1), we can plug in the values into the formula:
d = √((1 - (-4))^2 + (1 - 1)^2)
= √((1 + 4)^2 + 0^2)
= √(5^2)
= √25
= 5
Therefore, the shortest distance from Euclid Elementary School to Math Middle School is 5 miles.
Part B:
To find the shortest distance from Euclid Elementary School to Hypotenuse High School, we can again use the distance formula.
Using the coordinates of Euclid Elementary School as (-4, 1) and Hypotenuse High School as (-4, -3), we can plug in the values into the formula:
d = √((-3 - 1)^2 + (-4 - (-4))^2)
= √((-3 - 1)^2 + (-4 + 4)^2)
= √((-4)^2 + 0^2)
= √(16 + 0)
= √16
= 4
Therefore, the shortest distance from Euclid Elementary School to Hypotenuse High School is 4 miles.
Part C:
To find the shortest distance from Math Middle School to Hypotenuse High School, we can once again use the distance formula.
Using the coordinates of Math Middle School as (1, 1) and Hypotenuse High School as (-4, -3), we can plug in the values into the formula:
d = √((-3 - 1)^2 + (-4 - 1)^2)
= √((-3 - 1)^2 + (-4 - 1)^2)
= √((-4)^2 + (-5)^2)
= √(16 + 25)
= √41
Therefore, the shortest distance from Math Middle School to Hypotenuse High School is √41 miles.
Part D:
To determine who traveled the shortest distance, we can compare the distances traveled by Deon and Kaya.
Deon traveled from Hypotenuse High to Euclid Elementary and then to Math Middle School. The distance from Hypotenuse High to Euclid Elementary is 4 miles, and the distance from Euclid Elementary to Math Middle School is 5 miles. Therefore, the total distance traveled by Deon is 4 + 5 = 9 miles.
Kaya traveled from Hypotenuse High to Math Middle School along a straight path. From Part C, we know that the shortest distance from Math Middle School to Hypotenuse High School is √41 miles.
Comparing the two distances, we have 9 miles traveled by Deon versus √41 miles traveled by Kaya. Since √41 < 9, Kaya traveled the shortest distance.
Therefore, Kaya traveled the shortest distance.