Given that rectangle ABCD has coordinates A(0,0), B(12,0), C(12,6), and D(0,6), F is the midpoint of AB¯¯¯¯¯, and G is the midpoint of CD¯¯¯¯¯, which of the following proves that FB=GC?

1. By the Distance Formula, the coordinates of F are (6, 0) and the coordinates of G are (6, 6).
Then FB = 12, and GC = 12.
Thus FB = GC.

2. By the Distance Formula, the coordinates of F are (6, 12) and the coordinates of G are (6, 0).
Then FG = 6, and GC = 6.
Thus FG = GC.

3. By the Midpoint Formula, the coordinates of F are (6, 0) and the coordinates of G are (6, 6).
Then FB = 6, and GC = 6.
Thus FB = GC.

4. By the Midpoint Formula, the coordinates of F are (6, 6) and the coordinates of G are (6, 0).
Then FB = 6, and GC = 6.
Thus FB = GC.

The midpoint of the sides [tex]\overline{AB}[/tex] and [tex]\overline{CD}[/tex]  are the points that half way between  [tex]\overline{AB}[/tex] and [tex]\overline{CD}[/tex] respectively.

The option that proves [tex]\overline{FB}[/tex] = [tex]\overline{GC}[/tex] is option 3

Reasons:

The given coordinates of the vertices of rectangle ABCD are;

A(0, 0), B(12, 0), C(12, 6), and D(0, 6)

The midpoint of side [tex]\overline{AB}[/tex] = Point F

The midpoint of side [tex]\overline{CD}[/tex] = Point F

The coordinates of point F is given by the midpoint formula, as follows;

[tex]\displaystyle \mathrm{The \ midpoint \ of \ a \ line \ with \ endpoints \ (x_1, \ y_1) \ and \ (x_2, \ y_2) = \mathbf{ \left(\frac{x_1 + x_2}{2}, \ \frac{y_1 + y_2}{2} \right) }}[/tex]

Which gives;

[tex]\displaystyle \mathrm{The \ coordinate \ of \ F \ the \ midpoint \ of \ \overline{AB} = \mathbf{ \left(\frac{0 + 12}{2}, \ \frac{0 + 0}{2} \right)} =(6,\ 0)}[/tex]

[tex]\displaystyle \mathrm{The \ coordinate \ of \ G \ the \ midpoint \ of \ \overline{CD} = \left(\frac{12 + 0}{2}, \ \frac{6 + 6}{2} \right) =(6,\ 6)}[/tex]

[tex]\overline{FB}[/tex] is parallel to the x-axis, therefore;

Similarly, [tex]\overline{GC}[/tex] is parallel to the x-axis, therefore;

Which gives;

The correct option is that proves [tex]\overline{FB}[/tex] = [tex]\overline{GC}[/tex] is option 3.

By the Midpoint Formula, the coordinates of F are (6, 0) and the coordinates of G are (6, 6). Then [tex]\overline{FB}[/tex] = 6, and [tex]\overline{GC}[/tex] = 6. Thus [tex]\overline{FB}[/tex] = [tex]\overline{GC}[/tex]

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