Question

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

Explanation:

Step1: Recall the distance formula

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

Step2: Substitute the values into the formula

First, calculate \( x_2 - x_1 = 0-(-9)=0 + 9=9 \) and \( y_2 - y_1=-3 - 9=-12 \).
Then, substitute these into the distance formula: \( d=\sqrt{(9)^2+(-12)^2} \).

Step3: Simplify the expression inside the square root

Calculate \( 9^2 = 81 \) and \( (-12)^2=144 \). Then \( 81 + 144=225 \). So \( d=\sqrt{225} \)? Wait, no, wait, 81+144 is 225? Wait, 81 + 144: 80+140 = 220, 1+4=5, so 225. But wait, \(\sqrt{225} = 15\)? Wait, no, wait, did I make a mistake? Wait, \( x_2 - x_1=0 - (-9)=9 \), \( y_2 - y_1=-3 - 9=-12 \). Then \( (x_2 - x_1)^2=81 \), \( (y_2 - y_1)^2 = 144 \). Sum is 81 + 144 = 225. Then square root of 225 is 15. Wait, but the problem says "simplest radical form", but 225 is a perfect square. So the distance is 15. Wait, let me check again.

Wait, maybe I miscalculated the difference in y - coordinates. \( y_2 - y_1=-3 - 9=-12 \), squared is 144. \( x_2 - x_1=0 - (-9)=9 \), squared is 81. 81 + 144 = 225. Square root of 225 is 15. So the distance is 15.

Answer:

\( 15 \)