Question

question
find the distance between the two points in simplest radical form.
(-1, -1) and (5, -9)

Explanation:

Step1: Recall the distance formula

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

Step2: Identify the coordinates

Here, \( x_1=-1 \), \( y_1 = - 1 \), \( x_2=5 \), \( y_2=-9 \).

Step3: Substitute into the formula

First, calculate \( x_2 - x_1=5-(-1)=5 + 1=6 \) and \( y_2 - y_1=-9-(-1)=-9 + 1=-8 \).
Then, \( d=\sqrt{(6)^2+(-8)^2}=\sqrt{36 + 64}=\sqrt{100} \)? Wait, no, \( 36+64 = 100 \)? Wait, \( 6^2=36 \), \( (-8)^2 = 64 \), \( 36 + 64=100 \), but \( \sqrt{100}=10 \)? Wait, no, wait, let's check again. Wait, \( 5-(-1)=6 \), correct. \( -9-(-1)=-8 \), correct. Then \( 6^2 = 36 \), \( (-8)^2=64 \), sum is \( 36 + 64 = 100 \), square root of 100 is 10. Wait, but maybe I made a mistake? Wait, no, the distance formula is correct. Wait, let's recalculate \( y_2 - y_1 \): \( -9-(-1)=-9 + 1=-8 \), squared is 64. \( x_2 - x_1=5-(-1)=6 \), squared is 36. Sum is 100, square root is 10. So the distance is 10. Wait, but the problem says "simplest radical form", but 10 is an integer, which is a simplified radical (since \( \sqrt{100}=10 \)).

Answer:

The distance between the two points is \( \boldsymbol{10} \).