We are given point [tex]\( A \)[/tex] located at [tex]\((8,9)\)[/tex] and the distance from [tex]\( A \)[/tex] to another point [tex]\( B \)[/tex] is 10 units. We are asked to determine which of the provided equations could be used to calculate the coordinates of point [tex]\( B \)[/tex].
We start by understanding the distance formula between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex]:
[tex]\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
For our specific problem, the coordinates of point [tex]\( A \)[/tex] are [tex]\((8, 9)\)[/tex], and let's assume the coordinates of point [tex]\( B \)[/tex] are [tex]\((x, y)\)[/tex]. Using the distance formula, we have:
[tex]\[
10 = \sqrt{(x - 8)^2 + (y - 9)^2}
\][/tex]
Now let’s evaluate each of the given equations separately to see if they match this form or could be simplified/rearranged to match it.
1. Equation: [tex]\( 10 = \sqrt{(x + 8)^2 + (y + 9)^2} \)[/tex]
This equation suggests a different relationship. The term [tex]\( (x + 8) \)[/tex] implies that [tex]\( x \)[/tex] is 8 units to the left of the origin (assuming [tex]\( A \)[/tex] as the origin wouldn't make sense).
2. Equation: [tex]\( 10 = \sqrt{(x + 9)^2 + (y + 8)^2} \)[/tex]
This equation swaps the coordinates and similarly suggests a different relationship not stemming directly from point [tex]\( A \)[/tex].
3. Equation: [tex]\( 10 = \sqrt{(x - 8)^2 + (y - 9)^2} \)[/tex]
This matches directly the standard distance formula we derived and correctly uses the coordinates of point [tex]\( A \)[/tex]. This equation is correct and valid.
4. Equation: [tex]\( 10 = \sqrt{(x - 9)^2 + (y - 8)^2} \)[/tex]
This equation swaps the coordinates in a manner which is not related to point [tex]\( A \)[/tex]'s position properly.
After evaluating each equation, we find that only the third equation exactly matches what was originally derived from the distance formula while maintaining the relationship properly based on the coordinates of point [tex]\( A \)[/tex].
Therefore, the correct equation to calculate the coordinates for point [tex]\( B \)[/tex] is:
[tex]\[
\boxed{10 = \sqrt{(x - 8)^2 + (y - 9)^2}}
\][/tex]
