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geometry problem - please help!! [#permalink]
17 Feb 2006, 07:21

Hello,

Suppose you have an triangle in coordinate plane of whom you now only 2 points A and B, the third C is unknown.

Because you know point A and B you know the length between them.

Which additional information do you need to find the area of the triangle?

If we would know the length of |AC| and |BC|, the worst case, we can calculate the area.

But isn't it sufficient to know that the angle at the inner vertex of C is 90Â° ??

The height must be the same I guess, since the lenght of |AB| remains and there is only one possible length for the hieght. Am I wrong? Hope it's clear what I want.

Thanks

this is the original question

Attachments

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Last edited by allabout on 17 Feb 2006, 07:42, edited 2 times in total.

Re: geometry problem - please help!! [#permalink]
17 Feb 2006, 07:38

allabout wrote:

Hello,

Suppose you have an triangle in coordinate plane of whom you now only 2 points A and B, the third C is unknown.

Because you know point A and B you know their length.

Which additional information do you need to find the area of the triangle?

If we would know the length of |AC| and |BC|, the worst case, we can calculate the area.

But isn't it sufficient to know that the angle at the inner vertex of C is 90Â° ??

The height must be the same I guess, since the lenght of |AB| remains and there is only one possible length for the hieght. Am I wrong? Hope it's clear what I want.

Re: geometry problem - please help!! [#permalink]
17 Feb 2006, 08:11

allabout wrote:

Hello,

Suppose you have an triangle in coordinate plane of whom you now only 2 points A and B, the third C is unknown.

Because you know point A and B you know the length between them.

Which additional information do you need to find the area of the triangle?

If we would know the length of |AC| and |BC|, the worst case, we can calculate the area.

But isn't it sufficient to know that the angle at the inner vertex of C is 90Â° ??

The height must be the same I guess, since the lenght of |AB| remains and there is only one possible length for the hieght. Am I wrong? Hope it's clear what I want.

Thanks

this is the original question

allabout,

I agree with you. I would pick C too. _________________

"To dream anything that you want to dream, that is the beauty of the human mind. To do anything that you want to do, that is the strength of the human will. To trust yourself, to test your limits, that is the courage to succeed."

Now I understand that my suggestion was poorly worded. I'm trying to argue for D!

Given:

- Two points ( thus also the length between them)

Needed:

- either the length of the other two (statement 1)

- or we know that vertex at (x,y) is 90Â° ( I think the triangle is clearly determined if a side is given and the opposite angle is 90Â°.

Look at the OE please:
"From S1 it follows that y is either 3 or -1. In either case, the height of the triangle is 1. As the base of the triangle is 5, we can calculate the area.
S2 is insufficient. (x, y) can be close to the base (the area will be small) or far away from it (the area will be large)."

I think that if the height is determined because of the angle of 90Â°. The height can just be altered if the points "move together" or "apart". What am I thinking wrong?

Area of a triangle: sqrt[s(s-a)(s-b)(s-c)] where s= (a+b+c)/2

- or we know that vertex at (x,y) is 90Â° ( I think the triangle is clearly determined if a side is given and the opposite angle is 90Â°.

Here is my argument against this statement,

Imagine a circle with diameter AB, where A is (-2,2) and (3,2). In theory you could draw an infinite number of triangles with angle 90 inside the semicircle above or below (because angle inside the semicircle is always 90). Hence the solution is not unique. _________________

"To dream anything that you want to dream, that is the beauty of the human mind. To do anything that you want to do, that is the strength of the human will. To trust yourself, to test your limits, that is the courage to succeed."

- or we know that vertex at (x,y) is 90Â° ( I think the triangle is clearly determined if a side is given and the opposite angle is 90Â°.

Here is my argument against this statement,

Imagine a circle with diameter AB, where A is (-2,2) and (3,2). In theory you could draw an infinite number of triangles with angle 90 inside the semicircle above or below (because angle inside the semicircle is always 90). Hence the solution is not unique.

that's the point! thanks
I hope this won't happen to me again. Forget my message.

"To dream anything that you want to dream, that is the beauty of the human mind. To do anything that you want to do, that is the strength of the human will. To trust yourself, to test your limits, that is the courage to succeed."

My answer is B for the question. The reason is the following.

Pythogoran triplets are unique.
(3,4,5) (5,12,13).

In this case all we need to do is find the value of the sides of the triangle

The given sides (-2,2) and (3,2) form the base of the triagle as there y axis is the same. So these two points form the base of the triangle. The distance between the two points is 5.

Also given is the fact that the two lines of the triangle that meet at (x,y) has an angle 90 degrees. So these two lines cannot be the hypotenuse.

Hence the line joining (-2,2) and (3,2) is the hypotenuse.

from here we can conclude that the length of other two sides are 3 and 4.

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