Proving Congruent Sides in Quadrilateral ABCD Using Coordinate Geometry

In the field of geometry, it is often necessary to determine the properties of geometric figures using coordinates. This article will focus on the quadrilateral ABCD, with vertices A being at (-1, 3), B at (4, 4), C at (5, -3), and D at (-2, -2). We will use the distance formula from coordinate geometry to prove that ABCD has at least one pair of congruent sides.

Introduction to Coordinate Geometry

Coordinate geometry, also known as analytic geometry, combines the properties of algebra and geometry to solve problems. One of the most fundamental concepts in coordinate geometry is the distance formula, which calculates the distance between two points in a coordinate plane. The distance formula is given by:

[ PQ^2 (a - c)^2 (b - d)^2 ]

where (P) and (Q) are points with coordinates ((a, b)) and ((c, d)) respectively. This formula allows us to determine the length of any side of a geometric figure defined by coordinates.

Calculating the Length of Sides

To prove that quadrilateral ABCD has at least one pair of congruent sides, we will calculate the distance between each pair of adjacent vertices using the distance formula.

Length of AB

First, let's calculate the length of side AB, where A is at ((-1, 3)) and B is at ((4, 4)).

[ AB^2 (4 - (-1))^2 (4 - 3)^2 ][ AB^2 (4 1)^2 (4 - 3)^2 ][ AB^2 5^2 1^2 ][ AB^2 25 1 ][ AB^2 26 ][ AB sqrt{26} ]

Length of BC

Next, let's calculate the length of side BC, where B is at ((4, 4)) and C is at ((5, -3)).

[ BC^2 (5 - 4)^2 (-3 - 4)^2 ][ BC^2 1^2 (-7)^2 ][ BC^2 1 49 ][ BC^2 50 ][ BC sqrt{50} ]

Length of CD

Now, let's calculate the length of side CD, where C is at ((5, -3)) and D is at ((-2, -2)).

[ CD^2 (-2 - 5)^2 (-2 - (-3))^2 ][ CD^2 (-7)^2 (1)^2 ][ CD^2 49 1 ][ CD^2 50 ][ CD sqrt{50} ]

Length of AD

Finally, let's calculate the length of side AD, where A is at ((-1, 3)) and D is at ((-2, -2)).

[ AD^2 (-2 - (-1))^2 (-2 - 3)^2 ][ AD^2 (-2 1)^2 (-2 - 3)^2 ][ AD^2 (-1)^2 (-5)^2 ][ AD^2 1 25 ][ AD^2 26 ][ AD sqrt{26} ]

Conclusion

From our calculations, we can see that:

Thus, quadrilateral ABCD has at least one pair of congruent sides: AB and AD, as well as BC and CD, both of which have a length of (sqrt{26}) and (sqrt{50}) respectively.

By using coordinate geometry and the distance formula, we can effectively determine and prove properties of geometric figures based on their coordinates. This method is not only useful for identifying congruent sides but also for other geometric properties and relationships.