Question

find the distance between the two points in simplest radical form. (8, -4) and (4, 4)

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, \( x_1 = 8,y_1=- 4,x_2 = 4,y_2 = 4 \).

Step2: Substitute the values into the formula

Substitute the values into the formula:
\( d=\sqrt{(4 - 8)^2+(4-(-4))^2} \)
First, calculate the values inside the parentheses:
\( 4 - 8=-4 \) and \( 4-(-4)=4 + 4 = 8 \)
So the formula becomes \( d=\sqrt{(-4)^2+8^2} \)

Step3: Calculate the squares

Calculate \( (-4)^2=16 \) and \( 8^2 = 64 \)
Then \( d=\sqrt{16 + 64} \)

Step4: Add the numbers inside the square root

\( 16+64 = 80 \), so \( d=\sqrt{80} \)

Step5: Simplify the radical

Simplify \( \sqrt{80} \). We can factor 80 as \( 16\times5 \), and \( \sqrt{16\times5}=\sqrt{16}\times\sqrt{5}=4\sqrt{5} \)

Answer:

\( 4\sqrt{5} \)