Answer :
Sure! Let's determine the correct formula to find the distance from point [tex]\( D \)[/tex] to point [tex]\( A \)[/tex].
1. Understand the coordinates of the points:
- Point [tex]\( D \)[/tex] is at [tex]\( (0, b) \)[/tex].
- Point [tex]\( A \)[/tex] is at [tex]\( (0, 0) \)[/tex].
2. Apply the distance formula:
The distance formula between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
3. Substitute the coordinates of points [tex]\( D \)[/tex] and [tex]\( A \)[/tex] into the formula:
- [tex]\( x_1 = 0 \)[/tex], [tex]\( y_1 = 0 \)[/tex]
- [tex]\( x_2 = 0 \)[/tex], [tex]\( y_2 = b \)[/tex]
Plugging these values into the distance formula, we get:
[tex]\[ \text{Distance} = \sqrt{(0 - 0)^2 + (b - 0)^2} \][/tex]
Simplify the expression:
[tex]\[ \text{Distance} = \sqrt{0 + b^2} = \sqrt{b^2} = b \][/tex]
So, Martin can use the formula to determine the distance from point [tex]\( D \)[/tex] to point [tex]\( A \)[/tex]:
[tex]\[ \sqrt{(0-0)^2 + (b-0)^2} = \sqrt{b^2} = b \][/tex]
Examining the provided options:
- Option A: [tex]\((a-a)^2+(b-a)^2=b^2\)[/tex] is incorrect.
- Option B: [tex]\((0-0)^2+(b-0)^2=b^2\)[/tex] is correct numerically but does not include the square root.
- Option C: [tex]\(\sqrt{(a-a)^2+(b-0)^2}=\sqrt{b^2}=b\)[/tex] is incorrect in the x-component.
- Option D: [tex]\(\sqrt{(0-0)^2+(b-0)^2}=\sqrt{b^2}=b\)[/tex] is the correct option.
Therefore, the correct answer is:
[tex]\[ \boxed{\text{D}} \][/tex]
1. Understand the coordinates of the points:
- Point [tex]\( D \)[/tex] is at [tex]\( (0, b) \)[/tex].
- Point [tex]\( A \)[/tex] is at [tex]\( (0, 0) \)[/tex].
2. Apply the distance formula:
The distance formula between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
3. Substitute the coordinates of points [tex]\( D \)[/tex] and [tex]\( A \)[/tex] into the formula:
- [tex]\( x_1 = 0 \)[/tex], [tex]\( y_1 = 0 \)[/tex]
- [tex]\( x_2 = 0 \)[/tex], [tex]\( y_2 = b \)[/tex]
Plugging these values into the distance formula, we get:
[tex]\[ \text{Distance} = \sqrt{(0 - 0)^2 + (b - 0)^2} \][/tex]
Simplify the expression:
[tex]\[ \text{Distance} = \sqrt{0 + b^2} = \sqrt{b^2} = b \][/tex]
So, Martin can use the formula to determine the distance from point [tex]\( D \)[/tex] to point [tex]\( A \)[/tex]:
[tex]\[ \sqrt{(0-0)^2 + (b-0)^2} = \sqrt{b^2} = b \][/tex]
Examining the provided options:
- Option A: [tex]\((a-a)^2+(b-a)^2=b^2\)[/tex] is incorrect.
- Option B: [tex]\((0-0)^2+(b-0)^2=b^2\)[/tex] is correct numerically but does not include the square root.
- Option C: [tex]\(\sqrt{(a-a)^2+(b-0)^2}=\sqrt{b^2}=b\)[/tex] is incorrect in the x-component.
- Option D: [tex]\(\sqrt{(0-0)^2+(b-0)^2}=\sqrt{b^2}=b\)[/tex] is the correct option.
Therefore, the correct answer is:
[tex]\[ \boxed{\text{D}} \][/tex]
