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# Rhombus diagonals

## 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 of squares but not all of them because you could have a rhombus like this that comes in that where the angles aren't ninety 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 think about that a little bit clearer I'm gonna draw the diet I'm gonna draw the rhombus really it's kind of I'm gonna rotate a little bit so it looks a little bit like a diamond shape so but 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 it's 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 their 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 your ninth grade geometry you'd use the side-side-side theorem to prove that three sides or the 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 the shared side right over here over here will be congruent to the same the corresponding angle and the other triangle decide 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 that's the other base angle right this is an upside-down isosceles triangle this is a right side up one and so if this 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 you know I didn't actually 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 we take an altitude from each of these vertices down to this side right over here so an altitude by definition an altitude by definition is going to be perpendicular down here now and as I saw sleeze triangle is perfectly symmetrical if you drop an altitude from the I mean 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 each of each other you will also bisect you will also you will also bisect the opposite side this altitude is in fact a median of the triangle now we can do it on the other side we're getting the same exact thing is going to happen 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 can make the exact same argument over here you can think of a isosceles triangle 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 in the opposite side it's essentially a median of that triangle any isosceles triangle any isosceles triangle if that side is equal to that side if you drop an altitude these two triangles are going to be symmetric and you will have bisected you will have bisected the opposite side so by the same argument that side is equal to that side so the two diagonals of n rhombic of two of any rhombus are perpendicular to each other and they bisect each other and they bisect each other anyway hopefully you found that useful