3. Proving That a Quadrilateral Is a Parallelogram
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Find the slope of each side and compare it with the slope of its corresponding opposite side. Also, find and compare the midpoint of each diagonal.
See solution.
Let's show that the quadrilateral ABCD below is a parallelogram.
We will use two different methods – using the definition and using the Parallelogram Diagonals Converse.
Substitute ( -3,3) & ( 2,5)
| Endpoints | m = y_2-y_1/x_2-x_1 | Slope |
|---|---|---|
| B(2,5) and C(5,2) | m_2 = 2-5/5-2 | m_2 = -1 |
| C(5,2) and D(0,0) | m_3 = 0-2/0-5 | m_3 = 2/5 |
| D(0,0) and A(-3,3) | m_4 = 3-0/-3-0 | m_4 = -1 |
As we can see, m_1 = m_3 and m_2=m_4, which implies that AB∥ CD and BC∥ AD. Therefore, ABCD is a parallelogram.
In this part, we will draw the diagonals of the parallelogram.
Substitute ( -3,3) & ( 5,2)
Substitute ( 2,5) & ( 0,0)
Since the diagonals bisect each other, by the Parallelogram Diagonals Converse, we conclude that ABCD is a parallelogram.