Question

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.

Answer 1

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

The option that proves $\overline{FB}$ = $\overline{GC}$ is option 3

• 3. By the Midpoint Formula, the coordinates of F are (6, 0) and the coordinates of G are (6, 6). Then $\overline{FB}$ = 6, and $\overline{GC}$ = 6. Thus $\overline{FB}$ = $\overline{GC}$

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 $\overline{AB}$ = Point F

The midpoint of side $\overline{CD}$ = Point F

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

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

Which gives;

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

• The coordinates of point F = $\underline{(6, \ 0)}$

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

• The coordinates of point G = $\underline{(6, \ 6)}$

$\overline{FB}$ is parallel to the x-axis, therefore;

• The length of $\overline{FB}$ = 12 - 6 = 6

Similarly, $\overline{GC}$ is parallel to the x-axis, therefore;

• The length of $\overline{GC}$ = 12 - 6 = 6

Which gives;

• $\overline{FB}$ = $\overline{GC}$ = 6

The correct option is that proves $\overline{FB}$ = $\overline{GC}$ is option 3.

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

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