Question

the distance between two points is calculated as the root of the sum of the squares of the differences in their coordinates.
the form of a quadratic equation is used in geometry to derive the distance formula.
a crucial skill in using the distance formula effectively is manipulation.
the distance formula is often used in to design efficient layouts and ensure public amenities are accessible.
the of points is a basic concept in geometry that aids in understanding the distance formula.

Explanation:

1. The distance between two points is calculated as the square - root of the sum of the squares of the differences in their coordinates. This is the well - known distance formula in coordinate geometry.
2. The Pythagorean form of a quadratic equation (in a right - triangle context where the sides of the right - triangle are related to the coordinate differences) is used to derive the distance formula.
3. Algebraic manipulation is crucial to use the distance formula effectively, such as substituting values, simplifying expressions.
4. The distance formula is often used in urban planning to design efficient layouts and ensure public amenities are accessible.
5. The coordinates of points is a basic concept in geometry that helps in understanding the distance formula as the formula is based on the differences in coordinates of two points.

Answer:

1. square
2. Pythagorean
3. algebraic
4. urban planning
5. coordinates