Finding the Diagonals of Parallelograms, Rectangles, and Rhombi

Diagonals of Parallelograms, Rectangles, and Rhombi

All quadrilaterals have special traits. We’re going to look at the diagonals of parallelograms, rectangles, rhombi, and squares. Let’s start with the parallelogram. Now, all of these shapes are parallelograms, since a parallelogram is any 4-sided figure (a quadrilateral) with both pairs of opposites sides parallel, so that means that a rectangle is a parallelogram, and a rhombus, and a square.

Right now, we’re going to talk about just the general parallelogram, so on this parallelogram, ABCD, these diagonals would not be congruent to each other. If you’re looking at it, it’s like it’s leaning over, and if you think of a box, and you’re smushing the box down, then these corners A and C are going to be closer together than B and D, so that means the length of diagonal AC is shorter than the length of diagonal BD, so they’re not congruent.

However, the diagonals of all parallelograms do bisect each other, so that means that these segments are congruent and so are those segments. Again, since these are all parallelograms, that means that all of these diagonals of our parallelograms bisect each other, so we can do the same thing on all of our parallelograms.

Moving on to rectangles. Rectangles are special because they’re parallelograms that have all 4 angles, right angles, and that means that now our diagonals are congruent to each other, since these corners are the same distance away from each other.

We could say that segment EG is congruent to segment HF. A square is a parallelogram, but it’s also a rectangle, since a rectangle is just a parallelogram with 4 right angles, a square is a parallelogram with 4 right angles, so that means that it also has congruent diagonals, so we could say that diagonal MO, or segment MO, is congruent to segment PN.

Let’s talk about the rhombus now. A rhombus, again, is a parallelogram, it’s just a special type of parallelogram because all 4 of its sides are congruent. Since all 4 of our sides are congruent, then that means that our diagonals are perpendicular to each other, so all 4 of these angles around the center are all 90-degree angles, but the diagonals of the rhombus (like the parallelogram) are not necessarily congruent.

The only time that the diagonals of a rhombus would be congruent is if you were talking about a square, since a square is a rhombus, because a rhombus is a parallelogram with all 4 sides congruent. A square is also a parallelogram with all 4 sides congruent, therefore, a square is a rhombus.

Since it’s a rhombus it shares the same qualities as a rhombus does, so that means that its diagonals are perpendicular to each other, and, like we said earlier, the diagonals are congruent. That’s the only time that a rhombus would have congruent diagonals is if it were a square.

Let’s recap. In general, all parallelograms have diagonals that bisect each other, but only rectangles (and that includes squares because a square is a rectangle) have congruent diagonals, (besides trapezoids that we aren’t discussing right now) and rhombuses have perpendicular diagonals. That’s rhombuses, and then also squares, since squares are rhombuses.