Question

1. which pair of coordinates would have the longest distance between them based on the distance formula? a. (2,3) and (3,4) b. (1,1) and (2,2) c. (2,2) and (3,3) d. (1,1) and (3,4) 45. when a transversal cuts two lines and the alternate exterior angles are congruent, what can you conclude about the lines? a. the lines are skew. b. the lines are parallel. c. the lines are perpendicular. d. the lines are neither parallel nor perpendicular. 46. the distance between (10,15) and (2,9) is: a. 9 units b. 12 units c. 11 units d. 10 units

Explanation:

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

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

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.

For question 45:

Step1: Recall parallel - lines property

When a transversal cuts two lines and the alternate exterior angles are congruent, by the converse of the alternate - exterior angles theorem, the two lines are parallel. So the answer is b.

For question 46:

Step1: Apply distance formula

Let $(x_1,y_1)=(10,15)$ and $(x_2,y_2)=(2,9)$. Then $d=\sqrt{(2 - 10)^2+(9 - 15)^2}=\sqrt{(-8)^2+(-6)^2}=\sqrt{64 + 36}=\sqrt{100}=10$.

Answer:

1. D. $(1,1)$ and $(3,4)$
2. B. The lines are parallel.
3. D. 10 units