Question 31 of 40

Miguel 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]$A$[/tex] to point [tex]$B$[/tex]?

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



Answer :

To determine the distance between points [tex]\( A \)[/tex] and [tex]\( B \)[/tex] on a coordinate plane, we use the distance formula. The distance formula for two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:

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

In this specific problem, the points [tex]\( A \)[/tex] and [tex]\( B \)[/tex] have coordinates [tex]\( A(0, 0) \)[/tex] and [tex]\( B(a, 0) \)[/tex], respectively.

1. Identify the coordinates:
- [tex]\( A(0, 0) \)[/tex]
- [tex]\( B(a, 0) \)[/tex]

2. Substitute the coordinates into the distance formula:
[tex]\[ \sqrt{(a - 0)^2 + (0 - 0)^2} \][/tex]

3. Simplify the expression inside the square root:
[tex]\[ \sqrt{a^2 + 0} \][/tex]
[tex]\[ \sqrt{a^2} \][/tex]

4. Simplify further:
[tex]\[ a \][/tex]

The distance from point [tex]\( A \)[/tex] to point [tex]\( B \)[/tex] is [tex]\( a \)[/tex].

Review the provided options:

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

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

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

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

The correct formula and steps are reflected in option B:
[tex]\[ \sqrt{(a-0)^2+(0-0)^2}=\sqrt{a^2}=a \][/tex]

Thus, the answer is [tex]\( \boxed{B} \)[/tex].