Rhombus - Wikipedia
Learn more with some simple one-to-one relationship examples. Examples of one-to-one relationships include: A shark lives in one place (the ocean). Other examples include. people and places they What is good example of for loop in real life? Which of the following relations isn't a function, one-to-one, many-to-one or one-to-many? Tim Morgan, Ruby on Rails developer at Square . In plane Euclidean geometry, a rhombus (plural rhombi or rhombuses) is a simple The two diagonals of a rhombus are perpendicular; that is, a rhombus is an A rhombus therefore has all of the properties of a parallelogram: for example.
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.
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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.
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It bisects the opposite side. It's essentially a median of that triangle. Because area of ABCD-- actually let me write it this way. But since they're congruent, these two are going to be the same thing, so it's just going to be 2 times the area of ABC.
Now what is the area of ABC? What is the length of the base?
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Well the length of the base is AC. The base is AC. And then what is the height? What is the height here?
Well we know that this diagonal right over here, that it's a perpendicular bisector. So the height is just the distance from BE. So it's AC times BE, that is the height. This is an altitude. It intersects this base at a degree angle. So this is-- let me write it. So that's the area of just ABC, that's just the area of this broader triangle right up there, or that larger triangle right up there, that half of the rhombus.
But we decided that the area of the whole thing is two times that. So if we go back, if we use both this information and this information right over here, we have the area of ABCD is going to be equal to 2 times the area of ABC, where the area of ABC is this thing right over here. And then you see where this is going.
Fairly straightforward, which is a neat result. And actually, I haven't done this in a video. I'll do it in the next video.