Answered

Westonci.ca is the Q&A platform that connects you with experts who provide accurate and detailed answers. Join our Q&A platform to get precise answers from experts in diverse fields and enhance your understanding. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.

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]
Thanks for using our service. We aim to provide the most accurate answers for all your queries. Visit us again for more insights. Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Westonci.ca is here to provide the answers you seek. Return often for more expert solutions.