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Circles and Triangles

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Circles and Triangles

by anuptvm » Wed Dec 29, 2010 5:19 am
Which of the following lists the number of points at which a circle can intersect a triangle?

A. 2 and 6 only

B. 2, 4, and 6 only

C. 1, 2, 3, and 6 only

D. 1, 2, 3, 4 and 6 only

E. 1, 2, 3, 4, 5, and 6 only

My choice was B the OA is E

Could someone explain !
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by anshumishra » Wed Dec 29, 2010 5:36 am
anuptvm wrote:Which of the following lists the number of points at which a circle can intersect a triangle?

A. 2 and 6 only

B. 2, 4, and 6 only

C. 1, 2, 3, and 6 only

D. 1, 2, 3, 4 and 6 only

E. 1, 2, 3, 4, 5, and 6 only

My choice was B the OA is E

Could someone explain !
Answer is E.
One side of triangle can intersect the circle on two points (draw a chord and extend it as the side of the triangle and you will see) or touch the circle at one point if the side tangentially touches the circle or zero points otherwise.

There are a total of 3 sides, so it can intersect the circle on 1,2,3,4,5 and 6 points.
Let me show a few ->
1 point -> 1 side tangent , other side don't touch
2 points -> (1 side as extending chord, other sides don't touch) OR (Two sides as tangent and third side don't touch circle)
...........
...........

Hope that makes it clear.
Thanks
Anshu

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by anuptvm » Wed Dec 29, 2010 5:38 am
anshumishra wrote:
Answer is E.
One side of triangle can intersect the circle on two points (draw a chord and extend it as the side of the triangle and you will see) or touch the circle at one point if the side tangentially touches the circle or zero points otherwise.

There are a total of 3 sides, so it can intersect the circle on 1,2,3,4,5 and 6 points.
Let me show a few ->
1 point -> 1 side tangent , other side don't touch
2 points -> (1 side as extending chord, other sides don't touch) OR (Two sides as tangent and third side don't touch circle)
...........
...........

Hope that makes it clear.
Okay, but isn't intersection and touching the circle different? I counted only those scenarios were the side of the triangel cuts thru the circle ( 2 distinct points for each side)
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by anshumishra » Wed Dec 29, 2010 5:40 am
anuptvm wrote:
anshumishra wrote:
Answer is E.
One side of triangle can intersect the circle on two points (draw a chord and extend it as the side of the triangle and you will see) or touch the circle at one point if the side tangentially touches the circle or zero points otherwise.

There are a total of 3 sides, so it can intersect the circle on 1,2,3,4,5 and 6 points.
Let me show a few ->
1 point -> 1 side tangent , other side don't touch
2 points -> (1 side as extending chord, other sides don't touch) OR (Two sides as tangent and third side don't touch circle)
...........
...........

Hope that makes it clear.
Okay, but isn't intersection and touching the circle different? I counted only those scenarios were the side of the triangel cuts thru the circle ( 2 distinct points for each side)
No, touching means 1 point of intersection.
Thanks
Anshu

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by captcha » Wed Dec 29, 2010 6:16 am
IMO E
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by fskilnik@GMATH » Wed Dec 29, 2010 6:30 am
Hi there,

Anshu is correct: we are looking for 1 point in the intersection and that is THE SAME as AT LEAST 1 point in the intersection.

Please note that in Math the opposite (=negation) of "one ...." is "NONE ..." , therefore we were looking for cases (geometric scenarios) in which the intersection asked is non-empty, that is, in which there is (at least) one point in common to the circle and to the triangle.

Changing subject, IMHO people should not post "IMO ..." (at least) after the complete (potential) solution was presented... for 2 reasons:

(1) It is not a matter of opinion... we are discussing MATH here, not some pseudo-science theme...

(2) If someone

(a) thinks there is a flaw in another guy´s reasoning, please show it or ask about it ;
(b) did not understand another guy´s argument, please ask for more details.

If we respect that, the posts will be much more useful, don´t you all agree?

Regards to all, especially to captcha (no pun intended above, really)!
Fabio.
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by anshumishra » Wed Dec 29, 2010 6:38 am
fskilnik wrote:Hi there,

Anshu is correct: we are looking for 1 point in the intersection and that is THE SAME as AT LEAST 1 point in the intersection.

Please note that in Math the opposite (=negation) of "one ...." is "NONE ..." , therefore we were looking for cases (geometric scenarios) in which the intersection asked is non-empty, that is, in which there is (at least) one point in common to the circle and to the triangle.

Changing subject, IMHO people should not post "IMO ..." (at least) after the complete (potential) solution was presented... for 2 reasons:

(1) It is not a matter of opinion... we are discussing MATH here, not some pseudo-science theme...

(2) If someone

(a) thinks there is a flaw in another guy´s reasoning, please show it or ask about it ;
(b) did not understand another guy´s argument, please ask for more details.

If we respect that, the posts will be much more useful, don´t you all agree?

Regards to all, especially to captcha (no pun intended above, really)!
Fabio.
Thanks fsklink,

I can't agree with you more.
The only exception is when we are parallely working on a problem independently and post the solution within a few minutes interval. Then even if the answer and method match ( so in essence the solution doesn't add anything new) , no one can change that and we should live with that.
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by fskilnik@GMATH » Wed Dec 29, 2010 6:45 am
anshumishra wrote:The only exception is when we are parallely working on a problem independently and post the solution within a few minutes interval. Then even if the answer and method match ( so in essence the solution doesn't add anything new) , no one can change that and we should live with that.
Agreed, sure. Important: I have no authority nor the will to put people unconfortable... I just believe we should try to use arguments... it´s not voting, it´s solving/discussing problems, that´s all!

Cheers,
Fabio.
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