Question

which pair of coordinates would have the longest distance between them based on the distance formula?
a. (2,2) and (3,3)
b. (1,1) and (2,2)
c. (2,3) and (3,4)
d. (1,1) and (3,4)

what is the first step in deriving the distance formula from the pythagorean theorem?
a. solve for one variable
b. draw a circle
c. plot the points and form a right triangle
d. calculate the hypotenuse

which of the following is a practical application of the distance formula in astronomy?
a. calculating the distances between celestial bodies
b. counting the number of planets in a galaxy
c. measuring the size of the sun
d. determining the color of stars

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

Explanation:

First Question:

Step1: Recall distance formula

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$.

Step2: Calculate distance for option a

For points $(2,2)$ and $(3,3)$, $d_a=\sqrt{(3 - 2)^2+(3 - 2)^2}=\sqrt{1 + 1}=\sqrt{2}$.

Step3: Calculate distance for option b

For points $(1,1)$ and $(2,2)$, $d_b=\sqrt{(2 - 1)^2+(2 - 1)^2}=\sqrt{1+1}=\sqrt{2}$.

Step4: Calculate distance for option c

For points $(2,3)$ and $(3,4)$, $d_c=\sqrt{(3 - 2)^2+(4 - 3)^2}=\sqrt{1 + 1}=\sqrt{2}$.

Step5: Calculate distance for option d

For points $(1,1)$ and $(3,4)$, $d_d=\sqrt{(3 - 1)^2+(4 - 1)^2}=\sqrt{4 + 9}=\sqrt{13}$. Since $\sqrt{13}>\sqrt{2}$, option d has the longest distance.

Second Question:

The first step in deriving the distance formula from the Pythagorean theorem is to plot the points and form a right - triangle. So the answer is c.

Third Question:

The practical application of the distance formula in astronomy is calculating the distances between celestial bodies. So the answer is a.

Fourth Question:

In the derivation of the distance formula, the differences in coordinates like $x_2 - x_1$ and $y_2 - y_1$ are used to form the legs of the right - triangle. So the answer is likely a (assuming the option a is $x_2 - x_1$).

Answer:

1. D. $(1,1)$ and $(3,4)$
2. C. Plot the points and form a right triangle
3. A. Calculating the distances between celestial bodies
4. A. (assuming correct form of $x_2 - x_1$)