Apply the Pythagorean Theorem to find the length between ( 2 , 5 ) and ( 7 , 3 ) . Round your answer to the nearest hundredth, if necessary.(1 point)
To apply the Pythagorean Theorem to find the length between two points, we need to find the horizontal and vertical distances between the two points.
The horizontal distance is found by subtracting the x-coordinates of the two points.
Horizontal distance = 7 - 2 = 5
The vertical distance is found by subtracting the y-coordinates of the two points.
Vertical distance = 3 - 5 = -2
Now, we can apply the Pythagorean Theorem, which states that the square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the lengths of the other two sides.
Using the horizontal and vertical distances as the two sides of a right triangle, we have:
Hypotenuse^2 = Horizontal distance^2 + Vertical distance^2
Hypotenuse^2 = 5^2 + (-2)^2
Hypotenuse^2 = 25 + 4
Hypotenuse^2 = 29
To find the length of the hypotenuse, we take the square root of both sides:
Hypotenuse = sqrt(29)
Rounding to the nearest hundredth, Hypotenuse ≈ 5.39
Therefore, the length between (2, 5) and (7, 3) is approximately 5.39 units.
Apply the Pythagorean Theorem to find the length between ( − 1 , 6 ) and ( 5 , − 2 ) . Round your answer to the nearest hundredth, if necessary.(1 point)
To apply the Pythagorean Theorem to find the length between two points, we need to find the horizontal and vertical distances between the two points.
The horizontal distance is found by subtracting the x-coordinates of the two points.
Horizontal distance = 5 - (-1) = 6
The vertical distance is found by subtracting the y-coordinates of the two points.
Vertical distance = (-2) - 6 = -8
Now, we can apply the Pythagorean Theorem, which states that the square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the lengths of the other two sides.
Using the horizontal and vertical distances as the two sides of a right triangle, we have:
Hypotenuse^2 = Horizontal distance^2 + Vertical distance^2
Hypotenuse^2 = 6^2 + (-8)^2
Hypotenuse^2 = 36 + 64
Hypotenuse^2 = 100
To find the length of the hypotenuse, we take the square root of both sides:
Hypotenuse = sqrt(100)
Rounding to the nearest hundredth, Hypotenuse ≈ 10.00
Therefore, the length between (-1, 6) and (5, -2) is approximately 10.00 units.
To find the length between two points using the Pythagorean Theorem, we first need to determine the distance between the x-coordinates and the distance between the y-coordinates.
Given points:
Point 1: (2, 5)
Point 2: (7, 3)
Distance between x-coordinates:
x₁ = 2
x₂ = 7
Δx = x₂ - x₁ = 7 - 2 = 5
Distance between y-coordinates:
y₁ = 5
y₂ = 3
Δy = y₂ - y₁ = 3 - 5 = -2
The Pythagorean Theorem states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
To find the length between the two points, we can treat the Δx and Δy as the two sides of a right triangle, with the hypotenuse being the desired length.
Using the Pythagorean Theorem:
length² = (Δx)² + (Δy)²
length² = 5² + (-2)²
length² = 25 + 4
length² = 29
Taking the square root of both sides:
length = √29
Round the length to the nearest hundredth:
length ≈ 5.39
Therefore, the length between the points (2, 5) and (7, 3) is approximately 5.39 units.