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 states that the distance between two points with coordinates (x1, y1) and (x2, y2) is given by:
\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]
Let's calculate the distances for each part of the question.
Part A: Find the shortest distance, in miles, from Euclid Elementary School to Math Middle School.
The coordinates of Euclid Elementary School are (-4, 1) and the coordinates of Math Middle School are (1, 1). Plugging these values into the distance formula, we have:
\[d = \sqrt{(1 - (-4))^2 + (1 - 1)^2} = \sqrt{5^2 + 0^2} = \sqrt{25} = 5\]
So the shortest distance from Euclid Elementary School to Math Middle School is 5 miles.
Part B: Find the shortest distance, in miles, from Euclid Elementary School to Hypotenuse High School.
The coordinates of Euclid Elementary School are (-4, 1) and the coordinates of Hypotenuse High School are (-4, -3). Plugging these values into the distance formula, we have:
\[d = \sqrt{(-3 - 1)^2 + (-4 - (-4))^2} = \sqrt{(-4)^2 + (-3 - 1)^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5}\]
So the shortest distance from Euclid Elementary School to Hypotenuse High School is \(2\sqrt{5}\) miles.
Part C: Find the shortest distance, in miles, from Math Middle School to Hypotenuse High School.
The coordinates of Math Middle School are (1, 1) and the coordinates of Hypotenuse High School are (-4, -3). Plugging these values into the distance formula, we have:
\[d = \sqrt{(-3 - 1)^2 + (-4 - 1)^2} = \sqrt{(-4)^2 + (-5)^2} = \sqrt{16 + 25} = \sqrt{41}\]
So 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 answer this question, we need to compare the distances traveled by Deon and Kaya.
Deon traveled from Hypotenuse High to Euclid Elementary and then to Math Middle, covering the distances \(2\sqrt{5}\) miles and 5 miles. Therefore, Deon covered a total distance of \(2\sqrt{5} + 5\) miles.
Kaya traveled from Hypotenuse High to Math Middle along a straight path, covering the distance of \(\sqrt{41}\) miles.
Comparing both distances, we can see that \(2\sqrt{5} + 5\) is less than \(\sqrt{41}\). Therefore, Deon traveled the shortest distance.
could you try again bot
Apologies for the incorrect response. Let's try again.
Part A: To find the shortest distance from Euclid Elementary School to Math Middle School, we use the distance formula. The coordinates of Euclid Elementary School are (-4, 1), and the coordinates of Math Middle School are (1, 1).
Using the distance formula, the distance is given by:
\[d = \sqrt{(1 - (-4))^2 + (1 - 1)^2} = \sqrt{5^2 + 0^2} = \sqrt{25} = 5\]
So, the shortest distance from Euclid Elementary School to Math Middle School is 5 miles.
Part B: The coordinates of Euclid Elementary School are (-4, 1), and the coordinates of Hypotenuse High School are (-4, -3).
Using the distance formula, the distance is given by:
\[d = \sqrt{(-3 - 1)^2 + (-4 - (-4))^2} = \sqrt{(-4)^2 + (-3 - 1)^2} = \sqrt{16 + 16} = \sqrt{32}\]
So, the shortest distance from Euclid Elementary School to Hypotenuse High School is \(\sqrt{32}\) miles.
Part C: The coordinates of Math Middle School are (1, 1), and the coordinates of Hypotenuse High School are (-4, -3).
Using the distance formula, the distance is given by:
\[d = \sqrt{(-3 - 1)^2 + (-4 - 1)^2} = \sqrt{(-4)^2 + (-5)^2} = \sqrt{16 + 25} = \sqrt{41}\]
So, the shortest distance from Math Middle School to Hypotenuse High School is \(\sqrt{41}\) miles.
Part D: To determine who traveled the shortest distance, we compare the distances covered by Deon and Kaya.
Deon traveled from Hypotenuse High School to Euclid Elementary School, which is \(\sqrt{32}\) miles, and then from Euclid Elementary School to Math Middle School, which is 5 miles. The total distance traveled by Deon is \(\sqrt{32} + 5\) miles.
Kaya traveled directly from Hypotenuse High School to Math Middle School, which is \(\sqrt{41}\) miles.
Comparing the distances, we find that \(\sqrt{41}\) miles is less than \(\sqrt{32} + 5\) miles. Therefore, Kaya traveled the shortest distance.
could you not add extra numbers and letters such as sqrt, please?
Apologies for the confusion. Let's provide a simplified response using just the numerical values.
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 7 miles.
Part D: To determine who traveled the shortest distance, we compare the distances covered by Deon and Kaya.
Deon traveled from Hypotenuse High School to Euclid Elementary School, which is 4 miles, and then from Euclid Elementary School to Math Middle School, which is 5 miles. The total distance traveled by Deon is 9 miles.
Kaya traveled directly from Hypotenuse High School to Math Middle School, which is 7 miles.
Comparing the distances, we find that 7 miles is less than 9 miles. Therefore, Kaya traveled the shortest distance.
now show me the work for that answer without adding anything extra such as sqrt
Apologies for the confusion. Here's the revised answer without adding any additional expressions.
Part A:
The coordinates of Euclid Elementary School are (-4, 1), and the coordinates of Math Middle School are (1, 1).
To find the shortest distance between these points, we use the distance formula:
Distance = √[(x2 - x1)^2 + (y2 - y1)^2]
Plugging in the coordinates, we get:
Distance = √[(1 - (-4))^2 + (1 - 1)^2]
= √[(1 + 4)^2 + (0)^2]
= √[5^2 + 0^2]
= √25
= 5 miles
Part B:
The coordinates of Euclid Elementary School are (-4, 1), and the coordinates of Hypotenuse High School are (-4, -3).
Using the distance formula:
Distance = √[(x2 - x1)^2 + (y2 - y1)^2]
Plugging in the coordinates, we get:
Distance = √[(-3 - 1)^2 + (-4 - (-3))^2]
= √[(-4)^2 + (-7)^2]
= √[16 + 49]
= √65 miles
Part C:
The coordinates of Math Middle School are (1, 1), and the coordinates of Hypotenuse High School are (-4, -3).
Using the distance formula:
Distance = √[(x2 - x1)^2 + (y2 - y1)^2]
Plugging in the coordinates, we get:
Distance = √[(-3 - 1)^2 + (-4 - 1)^2]
= √[(-4)^2 + (-5)^2]
= √[16 + 25]
= √41 miles
Part D:
Deon traveled from Hypotenuse High School to Euclid Elementary School, which is 4 miles, and then from Euclid Elementary School to Math Middle School, which is 5 miles. The total distance traveled by Deon is 4 + 5 = 9 miles.
Kaya traveled directly from Hypotenuse High School to Math Middle School, which is 7 miles.
Comparing the distances, we find that 7 miles is less than 9 miles. Therefore, Kaya traveled the shortest distance.
could you organize it better please?
Certainly! Here's a revised and organized response:
Part A:
Coordinates of Euclid Elementary School: (-4, 1)
Coordinates of Math Middle School: (1, 1)
Using the distance formula:
Distance = √[(x2 - x1)^2 + (y2 - y1)^2]
Plugging in the coordinates:
Distance = √[(1 - (-4))^2 + (1 - 1)^2]
= √[(5)^2 + (0)^2]
= √[25 + 0]
= √25
= 5 miles
Part B:
Coordinates of Euclid Elementary School: (-4, 1)
Coordinates of Hypotenuse High School: (-4, -3)
Using the distance formula:
Distance = √[(x2 - x1)^2 + (y2 - y1)^2]
Plugging in the coordinates:
Distance = √[(-3 - 1)^2 + (-4 - (-3))^2]
= √[(-4)^2 + (-7)^2]
= √[16 + 49]
= √65 miles
Part C:
Coordinates of Math Middle School: (1, 1)
Coordinates of Hypotenuse High School: (-4, -3)
Using the distance formula:
Distance = √[(x2 - x1)^2 + (y2 - y1)^2]
Plugging in the coordinates:
Distance = √[(-3 - 1)^2 + (-4 - 1)^2]
= √[(-4)^2 + (-5)^2]
= √[16 + 25]
= √41 miles
Part D:
Deon's travel distance:
Hypotenuse High School to Euclid Elementary School: 4 miles
Euclid Elementary School to Math Middle School: 5 miles
Total distance = 4 + 5 = 9 miles
Kaya's travel distance:
Hypotenuse High School to Math Middle School: 7 miles
Comparing the distances, we find that Kaya traveled the shortest distance, which is 7 miles.