Rhombus Diagonals

Proof that the diagonals of a rhombus are perpendicular bisectors of each other

I'm going to do a quick argument or proof, as to why the diagonals of a rhombus are perpendicular.

So remember, a rhombus is just a parallelogram, where all four sides are equal.

In fact, if all four sides are equal, it has to be a parallelogram. Just to make things clear,

some rhombuses are squares, but not all of them, because you could have a rhombus like this,

that comes in where the angles aren't 90°. But squares are rhombuses, because all squares

they have 90° angles, that's not what makes them a rhombus, but all of the sides are equal.

So all squares are rhombuses, but not all rhombuses are squares. Now, with that said,

let's think about the diagonals of a rhombus, to think about that a little bit clearer,

I'm going to draw the dia- I'm going to draw the rhombus really as kind of,

I'm going to rotate it a little bit, so it looks a little bit like a diamond shape.

But notice, I'm not really changing the properties of the rhombus, I'm just changing

its orientation a little bit. I'm just changing its orientation.

So, a rhombus, by definition, the four sides are going to be equal. Now, let me draw one of its

diagonals, and the way I drew it right here is kind of a diamond.

One of its diagonals will be right along the horizontal, right like that.

Now this triangle on the top and the triangle on the bottom both share this side.

So that side is obviously is going to be the same length for both of these triangles.

In the other two sides of the triangles are also the same thing, they're sides of the actual rhombus.

So all three sides of this top triangle and the bottom triangle are the same.

So this top triangle and this bottom triangle are congruent.

They are congruent triangles. If you go back to your 9th grade Geometry unit, you'd use

the Side-Side-Side (SSS) theorem: if three sides are congruent, then the triangles themselves are congruent.

That also means all of the angles in the triangle are congruent. So the angle that is opposite this side,

this shared side, right over here will be congruent to the corresponding angle in the other triangle.

The angle opposite this side would be the same thing as that. Now, both of these triangles are also

isosceles triangles, so their base angles are going to be the same. So that's one base

angle, that's the other base angle. This is an upside down isosceles triangle,

this is a right-side-up one. And so, if these two are the same then these are also going to be the same.

They are going to be the same to each other, because this is an isosceles triangle.

And, they're also going to be the same to these other characters, down here, because these are

congruent triangles. Now, if we take an altitude... No, actually I don't have to talk about that

since, I don't think that will be relevant when we actually prove what we want to prove.

If we take an altitude from each of these vertices, down to this side, so an altitude by definition.

An altitude by definition is going to be perpendicular down here. Now, an isosceles triangle is perfectly

symmetrical. If you drop an altitude from the the top, or the unique angle, or the unique vertex

in an isosceles triangle you will split it into two symmetric right triangles. Two right triangles

that are essentially the mirror images of each other. You will also bisect the opposite side.

This altitude is, in fact, a median of the triangle. Now, we could do it on the other side,

the same exact thing is going to happen. We are bisecting this side over here, this is a right angle.

And so essentially, the combination of these two altitudes is really just a diagonal of

this rhombus, and it's at a right angle to the other diagonal of the rhombus.

And it bisects that other diagonal of the rhombus. We can make the exact same argument over here.

You could think of an isosceles triangle, over here. This is an altitude of it, it splits it into

two symmetric right triangles, it bisects the opposite side, it's essentially a median of that triangle.

Any isosceles triangle, of that side equal to that side, if you drop an altitude, these two triangles are

going to be symmetric, and you will have bisected the opposite side. So, by the same argument, that side

is equal to that side. So the two diagonals of any rhombus are perpendicular to each other

and they bisect each other. Anyway, hopefully you found that useful.