Miguel wants to use coordinate geometry to prove that the opposite sides of a rectangle are congruent. He places parallelogram [tex]$A B C D$[/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]$\sqrt{(a-0)^2+(0-0)^2}=\sqrt{a^2}=a$[/tex]

B. [tex]$(a-0)^2+(0-0)^2=a^2$[/tex]

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

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



Answer :

To determine the distance between two points in a coordinate plane, we use the distance formula:

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

In this problem, Miguel wants to find the distance between points [tex]\( A \)[/tex] and [tex]\( B \)[/tex]. The coordinates of point [tex]\( A \)[/tex] are [tex]\((0,0)\)[/tex] and the coordinates of point [tex]\( B \)[/tex] are [tex]\((a,0)\)[/tex].

Now, let's apply these coordinates to the distance formula:

- [tex]\( x_1 = 0 \)[/tex]
- [tex]\( y_1 = 0 \)[/tex]
- [tex]\( x_2 = a \)[/tex]
- [tex]\( y_2 = 0 \)[/tex]

Substitute the coordinates into the formula:

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

Simplify the expression inside the square root:

[tex]\[ \text{Distance} = \sqrt{a^2 + 0^2} \][/tex]

Since [tex]\( 0^2 = 0 \)[/tex], the expression simplifies to:

[tex]\[ \text{Distance} = \sqrt{a^2} \][/tex]

Finally, the square root of [tex]\( a^2 \)[/tex] is [tex]\( a \)[/tex] (assuming [tex]\( a \)[/tex] is non-negative):

[tex]\[ \text{Distance} = a \][/tex]

Thus, the formula Miguel can use to determine the distance from point [tex]\( A \)[/tex] to point [tex]\( B \)[/tex] is:

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

So, the correct option is:

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