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Proof that the diagonals of a rhombus are perpendicular bisectors of each other. Created by Sal Khan.
Video transcript
I want 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. And 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 degrees. But all squares are rhombuses, because all squares, they have 90-degree angles here. 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. And to think about that a little bit clearer, I'm going to draw the rhombus really as kind of-- I'm gonna rotate a little bit, so it looks a little bit like a diamond shape. So notice I'm not really changing any of the properties of the rhombus. I'm just drawing it, 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 going to be the same length for both of these triangles. And then 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 this 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 ninth grade geometry, you'd use the side-side-side theorem to prove that. Three sides are congruent, then the triangles themselves are congruent. But that also means that all 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. So it 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, right? 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're going to be the same to each other, because this is an isosceles triangle. And they're also going to be the same as these other two characters down here, because these are congruent triangles. Now, if we take an altitude, and actually, I didn't even have to talk about that, since actually, I don't think that'll be relevant when we actually want to prove what we want to prove. If we take an altitude from each of these vertices down to this side right over here. So 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-- I guess you could call it 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. And we could 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. Any isosceles triangle, if that side's 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's 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.