Question

the distance formula is derived from the theorem.
to find the distance between two points, you must calculate the square root of the sum of the squares of the differences in the and coordinates.
the coordinate system is essential for plotting points and applying the distance formula.
in a right triangle, the side opposite the right angle is known as the
engineers use the distance formula to ensure the of structures like bridges and buildings.
one application of the distance formula in biology is tracking the patterns of animals.
in mathematics, the application of the distance formula enhances the understanding of geometric using algebraic methods.
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:

Step1: Recall distance - formula origin

The distance formula is derived from the Pythagorean theorem.

Step2: Recall distance - formula calculation

To find the distance between two points, you calculate the square - root of the sum of the squares of the differences in the $x$ and $y$ coordinates.

Step3: Recall coordinate system

The Cartesian coordinate system is essential for plotting points and applying the distance formula.

Step4: Recall right - triangle terminology

In a right - triangle, the side opposite the right angle is the hypotenuse.

Step5: Recall engineering application

Engineers use the distance formula to ensure the stability of structures like bridges and buildings.

Step6: Recall biology application

One application of the distance formula in biology is tracking the movement patterns of animals.

Step7: Recall geometric understanding

In mathematics, the application of the distance formula enhances the understanding of geometric relationships using algebraic methods.

Step8: Recall distance - formula calculation

The distance between two points is calculated as the square root of the sum of the squares of the differences in their coordinates.

Step9: Recall quadratic - equation form

The standard form of a quadratic equation is used in geometry to derive the distance formula.

Step10: Recall skill for distance - formula

A crucial skill in using the distance formula effectively is algebraic manipulation.

Step11: Recall urban - planning application

The distance formula is often used in urban planning to design efficient layouts and ensure public amenities are accessible.

Step12: Recall geometric concept

The location of points is a basic concept in geometry that aids in understanding the distance formula.

Answer:

1. Pythagorean
2. $x$, $y$
3. Cartesian
4. hypotenuse
5. stability
6. movement
7. relationships
8. square
9. standard
10. algebraic
11. urban planning
12. location