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# Rhombus diagonals

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.