Let's go through the problem step by step to find the correct formula that could be used to calculate the coordinates of point [tex]\(B\)[/tex] given that point [tex]\(A\)[/tex] is located at [tex]\((6, 5)\)[/tex] and the distance from [tex]\(A\)[/tex] to [tex]\(B\)[/tex] is 5 units.
The distance formula in the coordinate plane is given by:
[tex]\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
Here, [tex]\((x_1, y_1)\)[/tex] represents the coordinates of point [tex]\(A\)[/tex] and [tex]\((x_2, y_2)\)[/tex] represents the coordinates of point [tex]\(B\)[/tex].
Given:
- The coordinates of point [tex]\(A\)[/tex] are [tex]\((6, 5)\)[/tex].
- The distance from [tex]\(A\)[/tex] to [tex]\(B\)[/tex] is 5 units.
We need to plug these values into the distance formula to find an equation involving [tex]\(x\)[/tex] and [tex]\(y\)[/tex] (the coordinates of point [tex]\(B\)[/tex]).
Substituting [tex]\( x_1 = 6 \)[/tex], [tex]\( y_1 = 5 \)[/tex], and [tex]\(\text{Distance} = 5\)[/tex], the formula becomes:
[tex]\[ 5 = \sqrt{(x - 6)^2 + (y - 5)^2} \][/tex]
This equation represents the relationship between the coordinates of point [tex]\(A\)[/tex] and point [tex]\(B\)[/tex] given the distance between them.
Let's compare this result with the given options:
1. [tex]\( 5 = \sqrt{(x - 6)^2 + (y - 5)^2} \)[/tex]
2. [tex]\( 5 = \sqrt{(x - 5)^2 + (y - 6)^2} \)[/tex]
3. [tex]\( 5 = \sqrt{(x + 6)^2 + (y + 5)^2} \)[/tex]
4. [tex]\( 5 = \sqrt{(x + 5)^2 + (y + 6)^2} \)[/tex]
The correct option that matches our derived formula is:
[tex]\[ 5 = \sqrt{(x - 6)^2 + (y - 5)^2} \][/tex]
So, the correct formula to calculate the coordinates for point [tex]\(B\)[/tex] is:
[tex]\[ 5 = \sqrt{(x - 6)^2 + (y - 5)^2} \][/tex]
