for this activity you will need to design a town with at least four different places, find the distance between each Place using the Pythagorean theorem or the distance formula, choose one row and calculate the distance using the alternative method did you get the same result?
Town Design:
Place 1: Town Hall
Place 2: Park
Place 3: School
Place 4: Supermarket
To calculate the distances between each place, we need to assign coordinates to their locations. Suppose the town is set on a 2-dimensional grid:
Town Hall (TH): (0, 0)
Park (P): (3, 0)
School (S): (0, 4)
Supermarket (SM): (2, 5)
Using these coordinates, we can calculate the distances between the places.
1. Distance between Town Hall and Park (THP):
Using the Pythagorean theorem:
THP = √[(3 - 0)^2 + (0 - 0)^2]
= √(9 + 0)
= √9
= 3 units.
2. Distance between Town Hall and School (THS):
Using the Pythagorean theorem:
THS = √[(0 - 0)^2 + (4 - 0)^2]
= √(0 + 16)
= √16
= 4 units.
3. Distance between Town Hall and Supermarket (THSM):
Using the Pythagorean theorem:
THSM = √[(2 - 0)^2 + (5 - 0)^2]
= √(4 + 25)
= √29 units (approx. 5.39 units).
4. Distance between Park and School (PS):
Using the Pythagorean theorem:
PS = √[(0 - 3)^2 + (4 - 0)^2]
= √(9 + 16)
= √25
= 5 units.
Alternative Method for Row 3 (Town Hall - Supermarket Distance - THSM):
Using the distance formula:
THSM = √[(2 - 0)^2 + (5 - 0)^2]
= √(4 + 25)
= √29 units (approx. 5.39 units).
The results obtained using both the Pythagorean theorem and the distance formula for the Town Hall - Supermarket distance (THSM) are the same.
thanks
You're welcome! If you have any more questions, feel free to ask. Happy designing!
To design a town with at least four different places, you can start by sketching out a simple map. Assign each place a location on the map, marking them with points or labels. For example, you can have Place A, Place B, Place C, and Place D.
To find the distance between each place using the Pythagorean theorem, you need to determine the coordinates of the points on the map. Let's say Place A is located at coordinates (x₁, y₁) and Place B is located at coordinates (x₂, y₂).
The Pythagorean theorem states that for any right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. In this case, the hypotenuse represents the straight line distance between the two places.
The distance between Place A and Place B can be calculated using the Pythagorean theorem as follows:
Distance = √((x₂ - x₁)² + (y₂ - y₁)²)
Alternatively, you can use the distance formula, which is a generalized version of the Pythagorean theorem. The distance formula states that the distance between two points in a coordinate plane can be found by taking the square root of the sum of the squares of the differences between their x and y coordinates.
Distance = √((x₂ - x₁)² + (y₂ - y₁)²)
Now, let's imagine you choose one row, such as the row containing Place A and Place B, and you calculate the distance using the alternative method. Suppose the coordinates are (x₁, y) for Place A and (x₂, y) for Place B.
Using the distance formula, the distance between Place A and Place B in this row would be:
Distance = √((x₂ - x₁)² + (y - y)²)
= √((x₂ - x₁)² + 0)
= √((x₂ - x₁)²)
= |x₂ - x₁|
Comparing this result to the distance calculated using the Pythagorean theorem, they should be the same because the y coordinates cancel out when calculating the difference.