Proving Quadrilateral ABCD is a Parallelogram

This article provides a comprehensive guide on how to prove that the quadrilateral ABCD with vertices A(0, 4), B(4, 2), C(2, -1), and D(-2, 1) is a parallelogram. We will explore multiple methods to verify this, including the properties of midpoints of diagonals and slope calculations.

The Method of Midpoints of Diagonals

To prove that a quadrilateral is a parallelogram using the midpoints of its diagonals, we start by determining the midpoints of the diagonals AC and BD.

Coordinates and Midpoints of Diagonals

Point x y A 0 4 B 4 2 C 2 -1 D -2 1

Diagonal AC:

The midpoint MAC is calculated as:

MAC (xA xC)/2, (yA yC)/2 (0 2)/2, (4 -1)/2 (2)/2, (3)/2 (1, 1.5)

Diagonal BD:

The midpoint MBD is calculated as:

MBD (xB xD)/2, (yB yD)/2 (4 -2)/2, (2 1)/2 (2)/2, (3)/2 (1, 1.5)

Comparing the midpoints, we find:

MAC (1, 1.5) and MBD (1, 1.5)

Since the midpoints of the diagonals AC and BD are the same, we can conclude that quadrilateral ABCD is a parallelogram.

Conclusion: Jorge’s statement that quadrilateral ABCD is a parallelogram is proven to be true.

The Method of Slope Calculation

An alternative method to prove that a quadrilateral is a parallelogram is by finding the slopes of its opposite sides. If both pairs of opposite sides have equal slopes, the quadrilateral is a parallelogram.

Slope of Opposite Sides

Slope of AB:

Slope(AB) (yB - yA) / (xB - xA) (2 - 4) / (4 - 0) -2 / 4 -0.5

Slope of DC:

Slope(DC) (yC - yD) / (xC - xD) (-1 - 1) / (2 - -2) -2 / 4 -0.5

Slope of AD:

Slope(AD) (yD - yA) / (xD - xA) (1 - 4) / (-2 - 0) -3 / -2 1.5

Slope of BC:

Slope(BC) (yC - yB) / (xC - xB) (-1 - 2) / (2 - 4) -3 / -2 1.5

Since the slopes of opposite sides are equal, quadrilateral ABCD is a parallelogram.

Conclusion: Based on the slope calculation, Jorge’s statement that quadrilateral ABCD is a parallelogram is proven to be true.

Conclusion

The quadrilateral ABCD is a parallelogram as verified by both the midpoint method of diagonals and the slope calculation method. Given the proven properties, Jorge’s statement is accurate.