To determine the distance between two points in a coordinate plane, we use the distance formula:
[tex]\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
In this problem, Miguel wants to find the distance between points [tex]\( A \)[/tex] and [tex]\( B \)[/tex]. The coordinates of point [tex]\( A \)[/tex] are [tex]\((0,0)\)[/tex] and the coordinates of point [tex]\( B \)[/tex] are [tex]\((a,0)\)[/tex].
Now, let's apply these coordinates to the distance formula:
- [tex]\( x_1 = 0 \)[/tex]
- [tex]\( y_1 = 0 \)[/tex]
- [tex]\( x_2 = a \)[/tex]
- [tex]\( y_2 = 0 \)[/tex]
Substitute the coordinates into the formula:
[tex]\[ \text{Distance} = \sqrt{(a - 0)^2 + (0 - 0)^2} \][/tex]
Simplify the expression inside the square root:
[tex]\[ \text{Distance} = \sqrt{a^2 + 0^2} \][/tex]
Since [tex]\( 0^2 = 0 \)[/tex], the expression simplifies to:
[tex]\[ \text{Distance} = \sqrt{a^2} \][/tex]
Finally, the square root of [tex]\( a^2 \)[/tex] is [tex]\( a \)[/tex] (assuming [tex]\( a \)[/tex] is non-negative):
[tex]\[ \text{Distance} = a \][/tex]
Thus, the formula Miguel can use to determine the distance from point [tex]\( A \)[/tex] to point [tex]\( B \)[/tex] is:
[tex]\[ \sqrt{(a-0)^2 + (0-0)^2} = \sqrt{a^2} = a \][/tex]
So, the correct option is:
A. [tex]\( \sqrt{(a-0)^2+(0-0)^2} = \sqrt{a^2} = a \)[/tex]
