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.

Explanation:

1. The distance formula between two - dimensional points $(x_1,y_1)$ and $(x_2,y_2)$ is $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$, which is derived from the Pythagorean theorem.
2. In the formula, we calculate the square - root of the sum of the squares of the differences in the $x$ and $y$ coordinates.
3. The Cartesian coordinate system is used for plotting points and applying the distance formula.
4. In a right - triangle, the side opposite the right angle is the hypotenuse.
5. Engineers use the distance formula to ensure the accuracy and stability of structures.
6. In biology, the distance formula can be used to track the movement patterns of animals.
7. In mathematics, the distance formula helps in understanding geometric relationships using algebraic methods.

Answer:

1. Pythagorean
2. $x$; $y$
3. Cartesian
4. hypotenuse
5. accuracy and stability
6. movement
7. relationships