Question

find the distance between the two points in simplest radical form. (9, -8) and (3, 0)

Explanation:

Step1: Recall distance formula

The distance \( d \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \).
Here, \( x_1 = 9 \), \( y_1 = -8 \), \( x_2 = 3 \), \( y_2 = 0 \).

Step2: Substitute values into formula

Calculate \( x_2 - x_1 = 3 - 9 = -6 \) and \( y_2 - y_1 = 0 - (-8) = 8 \).
Then, \( d = \sqrt{(-6)^2 + 8^2} \).

Step3: Simplify the expression

\( (-6)^2 = 36 \), \( 8^2 = 64 \). So, \( d = \sqrt{36 + 64} = \sqrt{100} \)? Wait, no, 36 + 64 is 100? Wait, 36 + 64 = 100? Wait, 36 + 64 is 100? Wait, no, 36 + 64 is 100? Wait, 6^2 is 36, 8^2 is 64, 36 + 64 = 100? Wait, no, 36 + 64 is 100? Wait, 10^2 is 100. Wait, but wait, let's recalculate: 3 - 9 is -6, squared is 36. 0 - (-8) is 8, squared is 64. 36 + 64 is 100? Wait, no, 36 + 64 is 100? Wait, 100 is 10 squared. Wait, but that would be 10, but let's check again. Wait, (9, -8) and (3, 0). The difference in x is 3 - 9 = -6, difference in y is 0 - (-8) = 8. Then squared terms: 36 and 64. Sum is 100. Square root of 100 is 10. Wait, but the problem says "simplest radical form". Wait, maybe I made a mistake. Wait, 36 + 64 is 100, which is a perfect square. So the distance is 10. Wait, but let's confirm the distance formula again. Yes, \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \). So substituting the values: \( x_1 = 9 \), \( y_1 = -8 \), \( x_2 = 3 \), \( y_2 = 0 \). So \( (3 - 9) = -6 \), squared is 36. \( (0 - (-8)) = 8 \), squared is 64. 36 + 64 = 100. Square root of 100 is 10. So the distance is 10.

Answer:

\( 10 \)