Answer:The distance formula is directly related to the Pythagorean Theorem.
The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. It can be written as:
c² = a² + b²
In this equation, c represents the length of the hypotenuse, and a and b represent the lengths of the other two sides.
Now, let's consider a coordinate plane with two points, (x₁, y₁) and (x₂, y₂). We can calculate the distance between these two points using the distance formula:
d = √((x₂ - x₁)² + (y₂ - y₁)²)
In this formula, d represents the distance between the two points.
To understand the connection between the distance formula and the Pythagorean Theorem, let's label the horizontal distance between the points as a, the vertical distance as b, and the distance between the points as c.
We can then rewrite the distance formula as:
d = √(a² + b²)
Comparing this equation with the Pythagorean Theorem, we can see that the distance formula is just a generalized version of the theorem. It calculates the distance between two points by finding the length of the hypotenuse in a right triangle formed by the horizontal and vertical distances.
In other words, the distance formula is an extension of the Pythagorean Theorem in the coordinate plane. By using the formula, we can calculate the distance between any two points, not just those forming right triangles.
For example, if we have two points (3, 4) and (6, 8), we can calculate the distance between them using the distance formula:
d = √((6 - 3)² + (8 - 4)²)
= √(3² + 4²)
= √(9 + 16)
= √25
= 5
So, the distance between the points (3, 4) and (6, 8) is 5 units.
In summary, the distance formula is related to the Pythagorean Theorem because it extends the theorem to calculate distances between any two points in a coordinate plane. By finding the length of the hypotenuse in a right triangle formed by the horizontal and vertical distances, we can determine the distance between the points.