Question

1. what is the euclidean distance from (2, 3) to the following points?

a. (9, 7)
b. (4, 8)
c. (2, 5.5)
d. (6, 0)
e. (1.1, 3)

Explanation:

Part a: Distance to (9, 7)

Step1: Recall Euclidean distance formula

The Euclidean distance \( d \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \). Here, \( (x_1, y_1) = (2, 3) \) and \( (x_2, y_2) = (9, 7) \).

Step2: Substitute values into formula

First, calculate \( x_2 - x_1 = 9 - 2 = 7 \) and \( y_2 - y_1 = 7 - 3 = 4 \). Then, \( d = \sqrt{7^2 + 4^2} = \sqrt{49 + 16} = \sqrt{65} \approx 8.06 \).

Step1: Apply Euclidean distance formula

Using \( (x_1, y_1) = (2, 3) \) and \( (x_2, y_2) = (4, 8) \), so \( x_2 - x_1 = 4 - 2 = 2 \) and \( y_2 - y_1 = 8 - 3 = 5 \).

Step2: Compute the distance

\( d = \sqrt{2^2 + 5^2} = \sqrt{4 + 25} = \sqrt{29} \approx 5.39 \).

Step1: Use distance formula

Here, \( x_2 - x_1 = 2 - 2 = 0 \) and \( y_2 - y_1 = 5.5 - 3 = 2.5 \).

Step2: Calculate distance

\( d = \sqrt{0^2 + 2.5^2} = \sqrt{0 + 6.25} = \sqrt{6.25} = 2.5 \).

Answer:

\( \sqrt{65} \) (or approximately 8.06)

Part b: Distance to (4, 8)