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.
Part B: Find the shortest distance, in miles, from Euclid Elementary School to Hypotenuse High School. Show every step of your work.
Part C: Find the shortest distance, in miles, from Math Middle School to Hypotenuse High School. Show every step of your work.
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.
To find the shortest distance between two points on a coordinate plane, we can use the distance formula. The distance formula is:
\(\text{Distance} = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\)
Let's calculate the distances for each part:
Part A: Distance between Euclid Elementary School and Math Middle School.
\(x_1 = -4\)
\(y_1 = 1\)
\(x_2 = 1\)
\(y_2 = 1\)
\(\text{Distance} = \sqrt{(1-(-4))^2+(1-1)^2}\)
\(\text{Distance} = \sqrt{5^2+0^2}\)
\(\text{Distance} = \sqrt{25}\)
\(\text{Distance} = 5\) miles
Therefore, the shortest distance from Euclid Elementary School to Math Middle School is 5 miles.
Part B: Distance between Euclid Elementary School and Hypotenuse High School.
\(x_1 = -4\)
\(y_1 = 1\)
\(x_2 = -4\)
\(y_2 = -3\)
\(\text{Distance} = \sqrt{(-3-1)^2+(-4-(-4))^2}\)
\(\text{Distance} = \sqrt{(-4)^2+(-3-1)^2}\)
\(\text{Distance} = \sqrt{16+4^2}\)
\(\text{Distance} = \sqrt{16+16}\)
\(\text{Distance} = \sqrt{32}\)
\(\text{Distance} = 4\sqrt{2}\) miles
Therefore, the shortest distance from Euclid Elementary School to Hypotenuse High School is \(4\sqrt{2}\) miles.
Part C: Distance between Math Middle School and Hypotenuse High School.
\(x_1 = 1\)
\(y_1 = 1\)
\(x_2 = -4\)
\(y_2 = -3\)
\(\text{Distance} = \sqrt{(-3-1)^2+(-4-1)^2}\)
\(\text{Distance} = \sqrt{(-4)^2+(-5)^2}\)
\(\text{Distance} = \sqrt{16+25}\)
\(\text{Distance} = \sqrt{41}\) miles
Therefore, the shortest distance from Math Middle School to Hypotenuse High School is \(\sqrt{41}\) miles.
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.
To determine who traveled the shortest distance, we need to compare Deon's total distance to Kaya's distance.
We calculated the following distances:
Deon: Hypotenuse High to Euclid Elementary = \(4\sqrt{2}\) miles
Euclid Elementary to Math Middle = 5 miles
Total distance for Deon = \(4\sqrt{2}+5\) miles
Kaya: Hypotenuse High to Math Middle = \(\sqrt{41}\) miles
Comparing the two distances, we can see that Kaya traveled the shortest distance, since \(\sqrt{41}<4\sqrt{2}+5\).
Therefore, Kaya traveled the shortest distance.