Ohio Assessments for Educators (OAE) Mathematics Practice Exam

What is the formula for calculating the distance between two points (x₁, y₁) and (x₂, y₂)?

1. √((x₂ - x₁)² + (y₂ - y₁)²)

2. (x₂ - x₁) + (y₂ - y₁)

3. (x₁ + x₂) + (y₁ + y₂)

4. |x₂ - x₁| + |y₂ - y₁|

The correct answer is: √((x₂ - x₁)² + (y₂ - y₁)²)

The formula for calculating the distance between two points \((x₁, y₁)\) and \((x₂, y₂)\) is derived from the Pythagorean theorem. This 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. In the context of the two points, the horizontal distance between them is \(|x₂ - x₁|\) and the vertical distance is \(|y₂ - y₁|\). If you visualize these points in a Cartesian coordinate system, they create a right triangle where these distances are the legs. The distance (the hypotenuse) can be found by taking the square root of the sum of the squares of these leg lengths. Thus, the formula becomes: \[ \text{Distance} = \sqrt{(x₂ - x₁)² + (y₂ - y₁)²} \] This formula calculates the Euclidean distance, which is the straight-line distance between the two points in a two-dimensional space. The other choices do not conform to this standard distance-calculation