Diagonals of Parallelograms

Parallelograms have many interesting properties. You might be aware that these shapes are quadrilaterals with four different sides. But if we look a little closer at these shapes, we see that the diagonals of these shapes are particularly interesting. But what's so special about the diagonals of a parallelogram, and what can they teach us about math? Let's find out.

Why the diagonals of a parallelogram are important

You might recall that when we talk about the diagonals of a parallelogram, we are talking about the lines that go from one corner to the opposite corner. Since a parallelogram has four corners, it also has two diagonals.

But here's the most important thing:

The diagonals bisect each other. While this might seem like a very simple observation, it can tell us some interesting things about the parallelogram. Let's take a closer look at the point at which the diagonals intersect.

This is the center of the parallelogram, and we can also call it the shape's "center of gravity". If we measure one diagonal line, then the point of intersection is the line's exact midpoint. This can help us make accurate observations and calculations in many situations.

If the diagonals bisect each other, then we know for sure that we're dealing with a parallelogram. But the opposite is also true: If we know that a quadrilateral is a parallelogram, we also know for sure that its diagonals must bisect each other.

Let's look at a picture of a parallelogram and its diagonals to make these rules easier to visualize:

Fun facts about parallelograms

Here are some fun facts about parallelograms:

The opposite angles are always equal
Two angles that lie on the same edge always add up to 180 degrees
A parallelogram can be a rectangle or a square
A rhombus is another example of a parallelogram
To find the area, multiply the base and the height $P = 2 (b + s)$
To find the perimeter, multiply the base and side length by 2

Formulas for finding the diagonal of a parallelogram

As you might have guessed, there's a formula for finding the diagonal of a parallelogram. After all, mathematicians have created formulas for just about everything in the universe!

First, we should point out that the two diagonals in a parallelogram are not equal. We can say that one diagonal is p while the second diagonal is q.

We know that there are four angles in a parallelogram -- but the opposite angles are equal. So we can simplify things by working with just two angles, A and B.

We also know that a parallelogram has two sets of equal sides, so we can use just two values to represent their lengths, x and y.

With these values, we can create the following formulas:

$p = \sqrt{x^{2} + y^{2} - 2 x y cos (A)} = \sqrt{x^{2} + y^{2} + 2 x y cos (B)}$

$q = \sqrt{x^{2} + y^{2} + 2 x y cos (A)} = \sqrt{x^{2} + y^{2} - 2 x y cos (B)}$

You might remember that "cos" stands for "cosine." Cosine is a ratio, and we find this ratio by dividing the side adjacent to the angle by the long side.

$p^{2} + q^{2} = 2 (x^{2} + y^{2})$

Topics related to the Diagonals of Parallelograms

Properties of Congruence

Parallel Lines

Polygon Exterior Angle Sum Theorem

Flashcards covering the Diagonals of Parallelograms

Advanced Geometry Flashcards

Common Core: High School - Geometry Flashcards

Practice tests covering the Diagonals of Parallelograms

Common Core: High School - Geometry Diagnostic Tests

Advanced Geometry Diagnostic Tests

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