Geometric Q w/ Lack of Data: "AB is hypotenuse of ABC, also biggest side of some..."

[shahar]

Geometric Q w/ Lack of Data: "AB is hypotenuse of ABC, also biggest side of some..."

"AB is the hypotenuse of ABC. AB is the biggest side of some quadriliteral."
(1) Suppose all the sides the quadriliteral are different from each other...
State True/Flase to the next statement: [and: Please explain to me: Why]
*The area of the triangle can be half of the area of the quadriletal.
*The quadriliteral can be a trapezoid.
*The ratio of the area between triangle and the quadriliteral can be a whole number.
(2) AB is still the biggest side, but the two of the side can be equal.
The same mission in (1).

Notice: There can be a lack of data but you have to fill the data so the statement will be as possible as a True Statement.

 

[HallsofIvy]

What "triangle" are the first and third statements referring to?

 

[Deleted member 4993]

"AB is the hypotenuse of ABC. AB is the biggest side of some quadriliteral."
(1) Suppose all the sides the quadriliteral are different from each other...
State True/Flase to the next statement: [and: Please explain to me: Why]
*The area of the triangle can be half of the area of the quadriletal.
*The quadriliteral can be a trapezoid.
*The ratio of the area between triangle and the quadriliteral can be a whole number.
(2) AB is still the biggest side, but the two of the side can be equal.
The same mission in (1).

Notice: There can be a lack of data but you have to fill the data so the statement will be as possible as a True Statement.

Exactly where are you stuck?
Where is your work/thought?

 

[shahar]

What "triangle" are the first and third statements referring to?

Any triangles that are can solve the question.
There question is "if it a possible to "draw" a triangle that as much as possible strict the description that described.

I do not have any work and I need clues how to begin - O.K...
What should I have to do to begin?

 

[Dr.Peterson]

"AB is the hypotenuse of ABC. AB is the biggest side of some quadriliteral."
(1) Suppose all the sides the quadriliteral are different from each other...
State True/Flase to the next statement: [and: Please explain to me: Why]
*The area of the triangle can be half of the area of the quadriletal.
*The quadriliteral can be a trapezoid.
*The ratio of the area between triangle and the quadriliteral can be a whole number.
(2) AB is still the biggest side, but the two of the side can be equal.
The same mission in (1).

Notice: There can be a lack of data but you have to fill the data so the statement will be as possible as a True Statement.

My first reaction is that this seems like a bizarre problem. I had to read carefully to see that there are some reasons the quadrilateral might be impossible.

Can you tell us the source and context of the question? If it is from a textbook, what topics are being discussed? If it is a contest problem, at what level?

As I understand it, you are given a right triangle ABC, and want to know if any quadrilateral fitting certain conditions exists (for any possible ABC). The triangle appears to affect the problem only via its area; so we can in part ignore the triangle itself and consider only a segment AB and a positive number K (which is that area, and therefore is less than a certain multiple of AB).

We are then asked whether a quadrilateral ABDE exists with AB as the longest side and all sides different, such that (a) its area is 2K; (b) it is a trapezoid; (c) its area is K/n. I think these are meant to be three separate questions, each answered true/false, not one combined condition, despite your singular "next statement". I also question whether (c) might be intended to be the ratio of the quadrilateral to the triangle, rather than vice versa -- "ratio between" can be unclear.

I would start with (2), which is less constrained. I can easily make a rectangle with twice the area of the triangle; would you consider AB the longest side if there are two sides of that length? I can also easily make a trapezoid with AB the longest side, since for (b) there is no restriction on area. And I think I can make a quadrilateral 1/2 the area of ABC.

But I'm not willing to put any more thought into it, because so much is unclear about the problem. I presume it is translated from another language; please check your translation and tell us anything you have to say about it, so we can get a better idea of what it really means.

 

[shahar]

O.K.

My first reaction is that this seems like a bizarre problem.

Thanks, it not only bizarre problem: When I read your reply. I check the question it is for very talented student. The question is not from a book, but from a "joke riddle" as "If you can solve, you can be Dr.". There is no answer, but when I look at the answer of the riddle, the answer was "Now, you a Doctor".
I waste our time with this riddle.
But thank to you, I can understand it.
How much time it take to you to solve it?
For me a two days, to figure.
Thank on your attention and answer.
I really think you very smart.
And it said: It for the Israeli SAT. (REALLY, I DO NOT THINK (!!!!!) THAT!).

 

[Dr.Peterson]

Thanks, it not only bizarre problem: When I read your reply. I check the question it is for very talented student. The question is not from a book, but from a "joke riddle" as "If you can solve, you can be Dr.". There is no answer, but when I look at the answer of the riddle, the answer was "Now, you a Doctor".
I waste our time with this riddle.
But thank to you, I can understand it.
How much time it take to you to solve it?
For me a two days, to figure.
Thank on your attention and answer.
I really think you very smart.
And it said: It for the Israeli SAT. (REALLY, I DO NOT THINK (!!!!!) THAT!).

If if it were stated more clearly, and I considered it worth pursuing, I could probably answer each part (by finding an example or proving impossible); but it would be a waste of time as it is.

Yes, sometimes the smartest thing to do is to decide not to do something.