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Nathan wants to use coordinate geometry to prove that the opposite sides of a rectangle are congruent. He places parallelogram [tex]\(ABCD\)[/tex] in the coordinate plane so that [tex]\(A\)[/tex] is [tex]\((0,0)\)[/tex], [tex]\(B\)[/tex] is [tex]\((a, 0)\)[/tex], [tex]\(C\)[/tex] is [tex]\((a, b)\)[/tex], and [tex]\(D\)[/tex] is [tex]\((0, b)\)[/tex].

What formula can he use to determine the distance from point [tex]\(C\)[/tex] to point [tex]\(D\)[/tex]?

A. [tex]\(\sqrt{(a-0)^2+(b-b)^2}=\sqrt{a^2}=a\)[/tex]

B. [tex]\(\sqrt{(a-0)^2+(0-0)^2}=\sqrt{a^2}=a\)[/tex]

C. [tex]\((a-0)^2+(b-b)^2=a^2\)[/tex]

D. [tex]\((a-0)^2+(0-0)^2=a^2\)[/tex]

Sagot :

GinnyAnswer
To determine the distance between two points in coordinate geometry, we use the distance formula. For two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex], the distance between them is given by:

[tex]\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]

Given the coordinates of the points [tex]\(C\)[/tex] and [tex]\(D\)[/tex]:
- [tex]\(C = (a, b)\)[/tex]
- [tex]\(D = (0, b)\)[/tex]

We can substitute these coordinates into the distance formula. The coordinates for [tex]\(C\)[/tex] are [tex]\((x_1, y_1) = (a, b)\)[/tex] and for [tex]\(D\)[/tex] are [tex]\((x_2, y_2) = (0, b)\)[/tex].

Now, apply the distance formula:

[tex]\[ \text{Distance from } C \text{ to } D = \sqrt{(0 - a)^2 + (b - b)^2} \][/tex]

Simplify the expression inside the square root:

[tex]\[ = \sqrt{(-a)^2 + (0)^2} \][/tex]
[tex]\[ = \sqrt{a^2 + 0} \][/tex]
[tex]\[ = \sqrt{a^2} \][/tex]
[tex]\[ = a \][/tex]

Therefore, the correct formula Nathan can use to determine the distance from point [tex]\(C\)[/tex] to point [tex]\(D\)[/tex] is:

[tex]\[ \sqrt{(a - 0)^2 + (b - b)^2} = \sqrt{a^2} = a \][/tex]

So the answer is:
A. [tex]\(\sqrt{(a-0)^2+(b-b)^2}=\sqrt{a^2}=a\)[/tex]
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