# Quadrilateral Family: In nice and green valley there is a building with 4 floors...

#### shahar

##### Junior Member
In nice and green valley there is a building with 4 floors.
In every floor there is a quadrilateral polygons, that is have a unique property.
Which quadrilateral polygons are in the building?

The answer is: Square, Rhombus, Trapezoid and Kite.

the answer of square I understood because a unique rectangle that has sizes that are equal.

Why the rest are living the building?

#### JeffM

##### Elite Member
In nice and green valley there is a building with 4 floors.
In every floor there is a quadrilateral polygons, that is have a unique property.
Which quadrilateral polygons are in the building?

The answer is: Square, Rhombus, Trapezoid and Kite.

the answer of square I understood because a unique rectangle that has sizes that are equal.

Why the rest are living the building?
Are you serious?

A rhombus is defined in geometery as a quadrilateral, all the sides of which are equal. A square is a rhombus that is also a rectangle.

A trapezoid is defined in geometry as a quadrilateral, with one pair of parallel sides.

I am not aware that "kite" has a technical definition in mathematics. The wing of a common kind of kite in the U.S. is a quadrilateral with one pair of touching sides of length x and another pair of touching sides of length y, where x > y. Thus, it is not a rhombus (all sides are not equal), nor is it a trapezoid (it has no parallel sides).

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#### shahar

##### Junior Member
What is kite

By Kite word, I meant to quadrilateral polygon that in Hebrew is called Dalton = two equal-sizes triangles (isosceles triangle) that have a shared side like the kite game.

What is the word of isosceles triangles that have a common side and create a quadrilateral?

#### JeffM

##### Elite Member
By Kite word, I meant to quadrilateral polygon that in Hebrew is called Dalton = two equal-sizes triangles (isosceles triangle) that have a shared side like the kite game.

What is the word of isosceles triangles that have a common side and create a quadrilateral?
There may be a word that means a quadrilateral formed by two non-congruent isosceles triangles with a common base, but I do not know it. I also tutor at English Learners Stack Exchange so if such a word exists in English, it is not common. Probably, I would say "kite-shaped quadrilateral," which would get across the idea of "dalton" to a mathematician who is a native English speaker.

I have edited my first post.

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#### Dr.Peterson

##### Elite Member
In nice and green valley there is a building with 4 floors.
In every floor there is a quadrilateral polygons, that is have a unique property.
Which quadrilateral polygons are in the building?

The answer is: Square, Rhombus, Trapezoid and Kite.

the answer of square I understood because a unique rectangle that has sizes that are equal.

Why the rest are living the building?
The word "kite", though non-technical, is easy enough to understand.

What I am unsure of is the meaning of "unique property". Can you explain what you think that means?

The whole question, of course, sounds silly; is it intended for children, perhaps? What is its context and source? Is there anything to suggest why they would give the answer they do?

The fact is that all the various kinds of quadrilateral are related, one being a special case of another. None are "unique" in an absolute sense, as I see it.

Your English grammar is often imperfect, as I'm sure you know; you mix up singular and plural, for example, which can make it hard to be sure what you mean, and here you have used the word "size" where I think you meant "side". I am wondering if that is happening here. Can you quote the problem word for word as given to you, in the original language, so we can judge its meaning better?

#### shahar

##### Junior Member
Here it is

השאלה היא:
בעמק יפה בין כרמים ושדות עומד בניין בן 4 קומות. חברו להם 4 מרובעים וביקשו להיות בו דיירים. התכנסו והעלו מחשבות כיצד יתפזרו בין הקומות. ואז עלה רעיון: זה שיהיה הראשון שיכריז על תכונה ייחודית - כל הקומה תהיה שלו.

התשובה היא: דלתון, טרפז, מעוין וריבוע .

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#### shahar

##### Junior Member
I don't know if the lanauge is O.K. so I use google translate

the question is:
In a beautiful valley between vineyards and fields stands a four-story building. They were joined by four quarters and asked to be tenants. They gathered and thought about how they would spread between the floors. And then an idea came up: it would be the first to declare a unique feature-the whole floor would be his.

The answer is: Dalton, trapeze, rhombus and square.

#### Dr.Peterson

##### Elite Member
In nice and green valley there is a building with 4 floors.
In every floor there is a quadrilateral polygons, that is have a unique property.
Which quadrilateral polygons are in the building?

The answer is: Square, Rhombus, Trapezoid and Kite.

the answer of square I understood because a unique rectangle that has sizes that are equal.

Why the rest are living the building?
the question is:
In a beautiful valley between vineyards and fields stands a four-story building. They were joined by four quarters and asked to be tenants. They gathered and thought about how they would spread between the floors. And then an idea came up: it would be the first to declare a unique feature-the whole floor would be his.

The answer is: Dalton, trapeze, rhombus and square.
Thanks. I, too, tried Google translate, and another site or two. Here is my attempt to put everything together to make a more meaningful version:

In a beautiful valley between vineyards and fields stands a four-story building. Four quadrilaterals came and asked to be tenants. They gathered and thought about how they would divide themselves between the floors. And then an idea came up: the first to declare a unique property, would get the whole floor.

The answer is: kite, trapezoid, rhombus and square.

Does that sound like a reasonable representation of the Hebrew?

I am thinking that perhaps it is the order of the four that is important, since as I interpret it, just the four came, and they needed to assign floors (rather than that four of all possible quadrilaterals are allowed in). So perhaps they are ordered by the number of properties they have. Another possibility, though, is that the answer is meant to be a list of categories into which everything would fit. (Neither of these ideas makes perfect sense to me.)

Clearly, though, the problem is still unclear, and is perhaps meant to be a riddle. But, also, a proper answer ought to include the reasons.

You haven't yet told us the context of the question. What type of source does it come from (e.g. children's book, contest problem, geometry textbook, ...)? Is there a picture with it? That may also help in interpreting it.

#### shahar

##### Junior Member
Here is the source

The flour need to accommodate by each 4-gon that have the unique properties that no others figure have.

In the article(now when I read it again), the 4-gons that I mentioned are the extreme figure in the hierarchy in the 4-gons family.
I repeat by the figures that in last in hierarchy.

(the hierarchy tree)
But why it so?!.... ?!

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#### JeffM

##### Elite Member
The flour need to accommodate by each 4-gon that have the unique properties that no others figure have.

In the article(now when I read it again), the 4-gons that I mentioned are the extreme figure in the hierarchy in the 4-gons family.
I repeat by the figures that in last in hierarchy.

(the hierarchy tree)
But why it so?!.... ?!
Because you do not give the hierarchy (and it is not yet clear to me that quadrilaterals can be organized into only one such tree), I am guessing.

But the "kite" has the attribute that it can be decomposed into two isosceles triangles, the trapezoid has the attribute that it has a pair of parallel sides, and the rhombus has the attribute that it can be decomposed into two congruent isosceles triangles and thus is a special kind of "kite." The square can be decomposed into two congruent right isosceles triangles, which makes it a special kind of rhombus (and so a very special kind of "kite"), but it is also a special kind of trapezoid because it has two pairs of sides that are parallel. So if I am guessing right, the square is at the "top of the tree."

#### Dr.Peterson

##### Elite Member
The flour need to accommodate by each 4-gon that have the unique properties that no others figure have.

In the article(now when I read it again), the 4-gons that I mentioned are the extreme figure in the hierarchy in the 4-gons family.
I repeat by the figures that in last in hierarchy.

(the hierarchy tree)
But why it so?!.... ?!
I don't think the figures you listed can be put in any one "correct" order. There are multiple ways to make a tree, depending on the properties you consider first. You could say that the square is a kind of rhombus, and the rhombus is a kind of kite (if you use an inclusive definition); and if you are defining trapezoid in the British sense of having no parallel sides, you can say that the kite is a kind of trapezoid. But that is not the order you say they gave.

If we focus on symmetry, the square has four lines of symmetry, the rhombus two, the kite one, and the trapezoid none, so this results in the same order.

I am wondering if it is really intended to be just a discussion-starter, rather than a problem with one correct answer. I couldn't translate the PDF you provided, but in part it looked like a discussion of teaching, which may have been focused on how to discuss such questions.

#### pka

##### Elite Member
The flour need to accommodate by each 4-gon that have the unique properties that no others figure have.
In the article(now when I read it again), the 4-gons that I mentioned are the extreme figure in the hierarchy in the 4-gons family.
I repeat by the figures that in last in hierarchy.
(the hierarchy tree) But why it so?!.... ?!
I find this to be a rather odd discussion. That is because the trapezoid seems to be the odd figure out.
Although the trapezoid is a quadrilateral like the other three the relation stops there.
For each of the kite, the rhombus , & the square all have two pair of congruent sides, they have perpendicular diagonals.
Moreover the square is a special case of rhombus.
Perhaps, if it were an isosceles trapezoid the problem is more coherent.

#### Denis

##### Senior Member
I'll take the one that's cheapest to rent

#### Jomo

##### Elite Member
I'll take the one that's cheapest to rent
I already told you that the cheapest rent for you is the (street) corner where you spend most of your time. Did you forget that already?
I'm ready to buy your house. Do you take e-bills? I have many of those.