Apply the Pythagorean Theorem to find the length between (−1, 6) and (5, −2) . Round your answer to the nearest hundredth, if necessary
We can use the Pythagorean Theorem to find the length between two points in a coordinate plane. The formula is:
c² = a² + b²
where c represents the hypotenuse of a right triangle and a and b represent the lengths of the other two sides.
In this case, the coordinates of the points are (-1, 6) and (5, -2). We can calculate the lengths of the two sides of the right triangle formed:
Side a = 5 - (-1) = 6
Side b = -2 - 6 = -8
Now we can use the Pythagorean Theorem to find the length of the hypotenuse (c):
c² = (6)² + (-8)²
c² = 36 + 64
c² = 100
Taking the square root of both sides, we find:
c ≈ √100
c ≈ 10
Therefore, the length between (-1, 6) and (5, -2) is approximately 10.
To apply the Pythagorean Theorem to find the distance between two points, we need to find the lengths of the horizontal and vertical sides of a right triangle formed by the two points.
Let's call the coordinates of the first point (x₁, y₁) and the coordinates of the second point (x₂, y₂).
Given:
Point 1: (-1, 6) --> (x₁ = -1, y₁ = 6)
Point 2: (5, -2) --> (x₂ = 5, y₂ = -2)
Using the Pythagorean Theorem, the distance (d) between the two points is calculated as:
d = √[(x₂ - x₁)² + (y₂ - y₁)²]
Substituting the given values:
d = √[(5 - (-1))² + (-2 - 6)²]
= √[6² + (-8)²]
= √[36 + 64]
= √100
= 10
Therefore, the distance between (-1, 6) and (5, -2) is 10.
To find the length between two points using the Pythagorean Theorem, you first need to determine the horizontal and vertical differences between the two points. Let's denote the coordinates of the first point (x1, y1) and the second point (x2, y2).
In this case, the coordinates of the first point are (-1, 6), and the coordinates of the second point are (5, -2).
To find the horizontal difference, subtract the x-coordinates:
Δx = x2 - x1 = 5 - (-1) = 6.
To find the vertical difference, subtract the y-coordinates:
Δy = y2 - y1 = -2 - 6 = -8.
Now, we can use the horizontal and vertical differences to apply the Pythagorean Theorem, which states that in a 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 horizontal difference (Δx) represents one side, the vertical difference (Δy) represents the other side, and the hypotenuse represents the distance between the two points.
Using the Pythagorean Theorem, the length between the two points can be calculated as follows:
Distance = √(Δx^2 + Δy^2) = √(6^2 + (-8)^2) = √(36 + 64) = √100 = 10.
Therefore, the length between the points (-1, 6) and (5, -2) is 10 units. Since rounding to the nearest hundredth is not necessary, the answer is already in whole units.