How can we show that a ray that begin in the center of a circle interest the circle by one point only?

shahar

shahar

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How can we show that a ray that begin in the center of a circle interest the circle by one point only?
 
This would depend on what you can consider as a given, i.e., your set of axioms.
The simplest proof that comes to my mind would be by contradiction: two points of intersection would have to have different distances from the center, which contradicts to the definition of a circle.
 
Here is the start of my proof.
A circle is the set of points a (finite) fixed distance from the center. A ray that starts at the center and goes in some direction for an infinite distance will clearly cross the circle in some point.
Now argue that this must be the only point! BIG Hint: Every point on this ray is a unique distance from the center.
 
blamocur said:
This would depend on what you can consider as a given, i.e., your set of axioms.
The simplest proof that comes to my mind would be by contradiction: two points of intersection would have to have different distances from the center, which contradicts to the definition of a circle.
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What type of axioms are you thinking about? I'm sure you're correct, but I just don't see what you are getting at.
 
Steven G said:
What type of axioms are you thinking about? I'm sure you're correct, but I just don't see what you are getting at.
Click to expand...
I did not think it through to reduce it all the way to some set of axioms. But I assumed as a given that if two points on a line are at the same distance from, and on the same side of another "center" point then those two points must be the same. I also assumed as a given that all points of a circle are at the same distance from the circle center.
 
Steven G said:
What type of axioms are you thinking about? I'm sure you're correct, but I just don't see what you are getting at.
Click to expand...
It seems to me that both of you are making essentially the same assumption: that each point on the ray is a different difference from its starting point:
blamocur said:
two points of intersection would have to have different distances from the center
Click to expand...
Steven G said:
Every point on this ray is a unique distance from the center.
Click to expand...

What we need to know is @shahar's context. He doesn't ask explicitly for a proof, so he may not have axioms in mind, but if there are some, he needs to state them. (I'm not sure what axiom systems typically say about rays; Euclid didn't use the concept, as I recall. But some form of "ruler axiom" seems appropriate.)
shahar said:
How can we show that a ray that begin in the center of a circle inter[sects] the circle [in] one point only?
Click to expand...
If "show" just means to informally convince you, then imagine you are moving along the ray: The distance from the center of the circle will be increasing. At only one point will this distance equal the radius of the circle.
 
Dr.Peterson said:
It seems to me that both of you are making essentially the same assumption: that each point on the ray is a different difference from its starting point:



What we need to know is @shahar's context. He doesn't ask explicitly for a proof, so he may not have axioms in mind, but if there are some, he needs to state them. (I'm not sure what axiom systems typically say about rays; Euclid didn't use the concept, as I recall. But some form of "ruler axiom" seems appropriate.)

If "show" just means to informally convince you, then imagine you are moving along the ray: The distance from the center of the circle will be increasing. At only one point will this distance equal the radius of the circle.
Click to expand...
After all, the purpose of the question to show one example of the condition that show it. In my opinion, this question should be written in first-grade pupils book but the source is high school pupil book. If one of you, the surfers of this website, wants a link to the source, tell me to put it here, in this thread.
 
shahar said:
After all, the purpose of the question to show one example of the condition that show it. In my opinion, this question should be written in first-grade pupils book but the source is high school pupil book. If one of you, the surfers of this website, wants a link to the source, tell me to put it here, in this thread.
Click to expand...
Can you at least give us an image of the actual question, and tell us the topic of the chapter it is in? Give us the link if you think it will help.

The question as you asked it said nothing about showing one example. If nothing else, I'd like to teach you how to ask clear questions, and one way is to quote exactly.
 
shahar said:
The link is

page 2, question 2 after the title תרגילים ומשימות
("exercises and tasks")
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According to Google Translate, the entire text of the exercise is:

2. Each ray, coming out of the center of the circle, cuts the the circle at one point.
Click to expand...
There is no actual question in the exercise, nor is there any indication in the header or other parts of the page which suggest that one is to "prove" the given statement. (In other words, this seems very poorly written.)
 
הראו שכל קרן, היוצאת ממרכז המעגל, חותכת את
המעגל בנקודה אחת.
Show that every ray coming out from the center of circle intersects the cicle at one point.
 
shahar said:
הראו שכל קרן, היוצאת ממרכז המעגל, חותכת את
המעגל בנקודה אחת.
Show that every ray coming out from the center of circle intersects the cicle at one point.
Click to expand...
Huh. Adding a space (which is not in the original text) in the middle of הראושכל, to get הראו שכל, changes the translation from "Every" to "Show that every". Thank you!
 
shahar said:
The link is

page 2, question 2 after the title תרגילים ומשימות
("exercises and tasks")
Click to expand...
I had Google translate the whole document. The problem says, "Show that every ray, emanating from the center of the circle, cuts the circle at one
point."

You'll have to tell us whether the word translated "show" can mean either a formal proof or an informal explanation; but the first page seems to suggest that actual proofs are in view (though perhaps not always stated as fully as we might).

But I can't judge from what I see how they might prove anything about a ray, since I see no facts about rays given.
 
Dr.Peterson said:
I had Google translate the whole document. The problem says, "Show that every ray, emanating from the center of the circle, cuts the circle at one
point."

You'll have to tell us whether the word translated "show" can mean either a formal proof or an informal explanation; but the first page seems to suggest that actual proofs are in view (though perhaps not always stated as fully as we might).

But I can't judge from what I see how they might prove anything about a ray, since I see no facts about rays given.
Click to expand...
The writer uses the word "show". I don't know if the writer of the question means to give an example or to give geometric proof.
 
shahar said:
The writer uses the word "show". I don't know if the writer of the question means to give an example or to give geometric proof.
Click to expand...
"Show that every ..." means prove, not give an example. If an example was requested, the word "every" would not have been used. Example means at least one.
 
shahar said:
The writer uses the word "show". I don't know if the writer of the question means to give an example or to give geometric proof.
Click to expand...
I noticed that the word for "show" there is different from the word used for "proof" on the first page; but that proves nothing. (I know only a little about ancient Hebrew, not modern.) I'd want to see if the book gives any worked examples using the word "show", which would indicate what they consider sufficient.

If this book doesn't make something clear, then you should either not use it, or contact the publisher and ask them to clarify. It doesn't help to ask us!

It is possible to use an example to "show" something is universally true (informally), by explaining the example appropriately (to show that the facts you are relying on will always be true). But it requires more than a mere example.
 
The answer:
1688230756062.png
תרגיל 2
הוכיחו שכל קרן היוצאת ממרכז המעגל חותכת את המעגל בנקודה אחת.

פתרון

כל נקודה על הקרן נמצאת במרחק מסוים מהמרכז
O,
ובכל מרחק נתון נמצאת נקודה אחת בלבד על הקרן.
הנקודה שמרחקה מהמרכז
R

נמצאת על המעגל, ואין נקודה אחרת שנמצאת באותו המרחק. מ.ש.ל.

QED=מ.ש.ל.
 
Google's translation, modified so it makes sense, is

Every point on the ray is a certain distance from the center O, and at any given distance there is only one point on the ray. The point that is distance R from the center is on the circle, and there is no other point that is at the same distance. QED.​

(I like that QED looks to Google like "mashal", which it translates as "parable". That's actually a word I recognize, as "proverb".)

Their "proof", while not referring specifically to axioms or theorems, is essentially what several of us said at the start:
Dr.Peterson said:
If "show" just means to informally convince you, then imagine you are moving along the ray: The distance from the center of the circle will be increasing. At only one point will this distance equal the radius of the circle.
Click to expand...
 
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