Certainly! Let's go through the steps of proving that the opposite sides of a parallelogram are of equal length.
### Definitions and Setup
Consider a parallelogram ABCD with vertices at:
- [tex]\( A(x_1, y_1) \)[/tex]
- [tex]\( B(x_2, y_2) \)[/tex]
- [tex]\( C(x_3, y_3) \)[/tex]
- [tex]\( D(x_4, y_4) \)[/tex]
In a parallelogram, by definition, the opposite sides are parallel. Specifically:
- [tex]\( AB \parallel CD \)[/tex]
- [tex]\( AD \parallel BC \)[/tex]
### Proving [tex]\( AB = CD \)[/tex]
1. Calculate the length of [tex]\( AB \)[/tex] and [tex]\( CD \)[/tex]:
The distance between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by the distance formula:
[tex]\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
So, the length of [tex]\( AB \)[/tex] is:
[tex]\[
AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
Similarly, the length of [tex]\( CD \)[/tex] is:
[tex]\[
CD = \sqrt{(x_4 - x_3)^2 + (y_4 - y_3)^2}
\][/tex]
2. Since [tex]\( AB \parallel CD \)[/tex] and [tex]\( ABCD \)[/tex] is a parallelogram, vectors [tex]\( \overrightarrow{AB} \)[/tex] and [tex]\( \overrightarrow{CD} \)[/tex] are:
[tex]\[
\overrightarrow{AB} = (x_2 - x_1, y_2 - y_1)
\][/tex]
[tex]\[
\overrightarrow{CD} = (x_4 - x_3, y_4 - y_3)
\][/tex]
3. The property of a parallelogram states that opposite sides are equal in magnitude:
[tex]\[
AB = CD
\][/tex]
Thus, we have proved [tex]\( AB = CD \)[/tex].
### Proving [tex]\( BC = AD \)[/tex]
1. Calculate the length of [tex]\( BC \)[/tex] and [tex]\( AD \)[/tex]:
Using the distance formula, the length of [tex]\( BC \)[/tex] is:
[tex]\[
BC = \sqrt{(x_3 - x_2)^2 + (y_3 - y_2)^2}
\][/tex]
The length of [tex]\( AD \)[/tex] is:
[tex]\[
AD = \sqrt{(x_1 - x_4)^2 + (y_1 - y_4)^2}
\][/tex]
2. Since [tex]\( AD \parallel BC \)[/tex] and [tex]\( ABCD \)[/tex] is a parallelogram, vectors [tex]\( \overrightarrow{BC} \)[/tex] and [tex]\( \overrightarrow{AD} \)[/tex] are:
[tex]\[
\overrightarrow{BC} = (x_3 - x_2, y_3 - y_2)
\][/tex]
[tex]\[
\overrightarrow{AD} = (x_1 - x_4, y_1 - y_4)
\][/tex]
3. Similarly, the property of a parallelogram states that opposite sides are equal in magnitude:
[tex]\[
BC = AD
\][/tex]
Thus, we have proved [tex]\( BC = AD \)[/tex].
### Conclusion
Therefore, by using the distance formula and the properties of parallelograms, we have shown that in a parallelogram [tex]\( ABCD \)[/tex]:
- [tex]\( AB = CD \)[/tex]
- [tex]\( BC = AD \)[/tex]
This means that the opposite sides of a parallelogram are indeed of equal length, as required.
