Question

1. which pair of coordinates would have the longest distance between them based on the distance formula?

a. (1,1) and (3,4)
b. (2,3) and (3,4)
c. (1,1) and (2,2)
d. (2,2) and (3,3)

Explanation:

Step1: Recall Distance Formula

The distance \( d \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \). We calculate the distance for each option.

Step2: Calculate Distance for Option a

For points \((1,1)\) and \((3,4)\):
\( x_1 = 1, y_1 = 1, x_2 = 3, y_2 = 4 \)
\( d = \sqrt{(3 - 1)^2 + (4 - 1)^2} = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13} \approx 3.605 \)

Step3: Calculate Distance for Option b

For points \((2,3)\) and \((3,4)\):
\( x_1 = 2, y_1 = 3, x_2 = 3, y_2 = 4 \)
\( d = \sqrt{(3 - 2)^2 + (4 - 3)^2} = \sqrt{1^2 + 1^2} = \sqrt{1 + 1} = \sqrt{2} \approx 1.414 \)

Step4: Calculate Distance for Option c

For points \((1,1)\) and \((2,2)\):
\( x_1 = 1, y_1 = 1, x_2 = 2, y_2 = 2 \)
\( d = \sqrt{(2 - 1)^2 + (2 - 1)^2} = \sqrt{1^2 + 1^2} = \sqrt{2} \approx 1.414 \)

Step5: Calculate Distance for Option d

For points \((2,2)\) and \((3,3)\):
\( x_1 = 2, y_1 = 2, x_2 = 3, y_2 = 3 \)
\( d = \sqrt{(3 - 2)^2 + (3 - 2)^2} = \sqrt{1^2 + 1^2} = \sqrt{2} \approx 1.414 \)

Answer:

a. (1,1) and (3,4)