Question

find the distance between the two points in simplest radical form.

Explanation:

Step1: Identify coordinates

First, find the coordinates of the two points. The first point (let's say \( A \)) is at \( (1, 8) \) (since it's 1 unit right on the x - axis and 8 units up on the y - axis). The second point (let's say \( B \)) is at \( (5, - 4) \) (5 units right on the x - axis and 4 units down on the y - axis, so y - coordinate is - 4).

Step2: Apply distance formula

The distance formula between two points \( (x_1,y_1) \) and \( (x_2,y_2) \) is \( d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2} \).
Substitute \( x_1 = 1,y_1 = 8,x_2 = 5,y_2=-4 \) into the formula:
\( x_2 - x_1=5 - 1 = 4 \)
\( y_2 - y_1=-4 - 8=-12 \)
Then \( d=\sqrt{(4)^2+(-12)^2}=\sqrt{16 + 144}=\sqrt{160} \)

Step3: Simplify the radical

Simplify \( \sqrt{160} \). We know that \( 160=16\times10 \), so \( \sqrt{160}=\sqrt{16\times10}=\sqrt{16}\times\sqrt{10}=4\sqrt{10} \)

Answer:

\( 4\sqrt{10} \)