To determine the distance between two points in the coordinate plane, Martin can use the distance formula. The distance formula between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
In this problem, point [tex]\(D\)[/tex] has coordinates [tex]\((0, b)\)[/tex] and point [tex]\(A\)[/tex] has coordinates [tex]\((0, 0)\)[/tex].
Let's apply the distance formula step-by-step:
1. Identify the coordinates of points [tex]\(D\)[/tex] and [tex]\(A\)[/tex]:
- Coordinates of [tex]\(D\)[/tex]: [tex]\( (0, b) \)[/tex]
- Coordinates of [tex]\(A\)[/tex]: [tex]\( (0, 0) \)[/tex]
2. Substitute the coordinates into the distance formula:
- [tex]\( x_1 = 0 \)[/tex]
- [tex]\( y_1 = b \)[/tex]
- [tex]\( x_2 = 0 \)[/tex]
- [tex]\( y_2 = 0 \)[/tex]
[tex]\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
3. Substitute the specific coordinates into the formula:
[tex]\[
\text{Distance} = \sqrt{(0 - 0)^2 + (b - 0)^2}
\][/tex]
4. Simplify the expression inside the square root:
[tex]\[
\text{Distance} = \sqrt{0 + b^2}
\][/tex]
5. Simplify further:
[tex]\[
\text{Distance} = \sqrt{b^2}
\][/tex]
6. Since the square root of [tex]\(b^2\)[/tex] is [tex]\(b\)[/tex] (assuming [tex]\(b \geq 0\)[/tex]):
[tex]\[
\text{Distance} = b
\][/tex]
Therefore, the correct formula Martin can use to determine the distance from point [tex]\(D\)[/tex] to point [tex]\(A\)[/tex] is:
[tex]\[
\sqrt{(0-0)^2+(b-0)^2}=\sqrt{b^2}=b
\][/tex]
Hence, the correct answer is:
A. [tex]\(\sqrt{(0-0)^2+(b-0)^2}=\sqrt{b^2}=b\)[/tex]
