diagonals of a parallelogram — Definition, Formula & Examples
The diagonals of a parallelogram are the two segments connecting opposite vertices, and their key property is that they bisect each other — each diagonal cuts the other into two equal halves at the point where they cross.
In a parallelogram , diagonals and intersect at a point such that and . That is, is the midpoint of both diagonals.
Key Formula
Where:
- = Vertices of the parallelogram taken in order
- = Intersection point (midpoint) of both diagonals
How It Works
To use the bisection property, find where the two diagonals intersect; that point is the midpoint of each diagonal. If you know the coordinates of two opposite vertices, you can compute the midpoint to locate the intersection. Conversely, if you know one vertex and the intersection point, you can find the opposite vertex by reflecting through the midpoint. This property holds for every parallelogram — whether it is a rectangle, rhombus, square, or a general slanted parallelogram.
Worked Example
Problem: Parallelogram ABCD has vertices A(1, 2), B(5, 2), C(7, 6), and D(3, 6). Find the point where the diagonals intersect and verify the bisection property.
Midpoint of AC: Average the coordinates of A and C.
Midpoint of BD: Average the coordinates of B and D.
Verify: Both midpoints are the same, confirming the diagonals bisect each other at (4, 4).
Answer: The diagonals intersect at (4, 4), the midpoint of each diagonal.
Why It Matters
The bisection property is one of the standard tests used in coordinate geometry proofs: if you show that a quadrilateral's diagonals share a midpoint, you have proved it is a parallelogram. This appears regularly on SAT, ACT, and high-school geometry exams when you need to classify quadrilaterals or find unknown vertices.
Common Mistakes
Mistake: Assuming the diagonals are equal in length.
Correction: Diagonals of a general parallelogram bisect each other but are not necessarily equal. Equal diagonals occur only in rectangles (and squares).