An amusement park has three roller coasters, the “Falcon Flyer” placed at (50, 300), the “Sprinter” placed at (400, 550), and the “Air Rider” placed at (800, 250). All measurements are in meters. The roller coasters are placed in positions to form a right triangle. true or false?
False.
To determine whether the roller coasters are placed in positions to form a right triangle, we need to calculate the distances between the coaster positions. The positions given are the coordinates (x, y) of each coaster.
Using the distance formula, the distance between two points (x1, y1) and (x2, y2) is given by:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Let's calculate the distances:
Distance between Falcon Flyer and Sprinter:
d1 = sqrt((400 - 50)^2 + (550 - 300)^2)
= sqrt(350^2 + 250^2)
= sqrt(122500 + 62500)
= sqrt(185000)
≈ 430.116
Distance between Falcon Flyer and Air Rider:
d2 = sqrt((800 - 50)^2 + (250 - 300)^2)
= sqrt(750^2 + (-50)^2)
= sqrt(562500 + 2500)
= sqrt(565000)
≈ 751.153
Distance between Sprinter and Air Rider:
d3 = sqrt((800 - 400)^2 + (250 - 550)^2)
= sqrt(400^2 + (-300)^2)
= sqrt(160000 + 90000)
= sqrt(250000)
= 500
Since the square of the longest side (d2) is not equal to the sum of the squares of the other two sides (d1^2 + d3^2), the triangle formed by the roller coasters is not a right triangle. Therefore, the statement is False.
To determine if the roller coasters are placed in positions to form a right triangle, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
Let's calculate the distances between the roller coasters and see if the lengths satisfy the Pythagorean theorem:
Distance between Falcon Flyer and Sprinter:
x₁ = 400 - 50 = 350
y₁ = 550 - 300 = 250
Distance = √(x₁² + y₁²) = √(350² + 250²)
Distance between Falcon Flyer and Air Rider:
x₂ = 800 - 50 = 750
y₂ = 250 - 300 = -50
Distance = √(x₂² + y₂²) = √(750² + (-50)²)
Distance between Sprinter and Air Rider:
x₃ = 800 - 400 = 400
y₃ = 250 - 550 = -300
Distance = √(x₃² + y₃²) = √(400² + (-300)²)
Now, we compare the squares of the distances:
Distance² between Falcon Flyer and Sprinter = (350² + 250²)
Distance² between Falcon Flyer and Air Rider = (750² + (-50)²)
Distance² between Sprinter and Air Rider = (400² + (-300)²)
If the square of the distance between Falcon Flyer and Sprinter is equal to the sum of the squares of the distances between Falcon Flyer and Air Rider and Sprinter and Air Rider, then the roller coasters are placed in positions to form a right triangle.
So, to determine if the statement is true or false, we need to compare the above values.
using the distance formula , find the distances between pairs of coasters
do the distances fit Pythagoras' formula ... a^2 + b^2 = c^2