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 \( d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2} \). Here, \( x_1=-9,y_1 = 9,x_2=0,y_2=-3 \).

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 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 \( 81+144 = 225 \), square root of 225 is 15. Wait, but 225 is a perfect square. So the distance is 15. Wait, but let me check again. The distance formula is correct. So the steps are:

1. Identify \( x_1,y_1,x_2,y_2 \): \( x_1=-9,y_1 = 9,x_2=0,y_2=-3 \)
2. Apply distance formula: \( d=\sqrt{(0 - (-9))^2+(-3 - 9)^2}=\sqrt{(9)^2+(-12)^2} \)
3. Calculate squares: \( 9^2 = 81 \), \( (-12)^2=144 \)
4. Sum: \( 81 + 144=225 \)
5. Take square root: \( \sqrt{225}=15 \)

Wait, but 225 is a perfect square, so the simplest radical form is 15.

Answer:

15