Question

which theorem is the basis for deriving the distance formula?
a. the triangle inequality theorem
b. the pythagorean theorem
c. the midpoint theorem
d. the circle theorem

for which purpose is the distance formula particularly important in robotics?
a. signal processing
b. navigating between points
c. power management
d. sensor tuning

which coordinate difference is used to form one leg of the triangle in the distance formula derivation?
a. $y_2 - y_1$
b. $x_2 + x_1$
c. $y_2 + y_1$
d. $x_2 - x_1$

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 $a^{2}+b^{2}=c^{2}$, where $a=\vert x_2 - x_1\vert$ and $b=\vert y_2 - y_1\vert$ are the legs of a right - triangle and $c$ is the hypotenuse (the distance between the two points).
2. In robotics, the distance formula is crucial for navigating between points. Robots need to calculate the distance between their current position and a target position to plan their movement paths.
3. When deriving the distance formula for two points $(x_1,y_1)$ and $(x_2,y_2)$ in a coordinate plane, the differences $x_2 - x_1$ and $y_2 - y_1$ form the legs of the right - triangle used in the Pythagorean - based derivation.

Answer:

1. b. The Pythagorean Theorem
2. b. Navigating between points
3. d. $x_2 - x_1$