interception points of circle and rectangle

shahar

shahar

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The figure ABCD is a rectangle. E is the midpoint of the side AD.
Which answer from the following numbers cannot be the number of points of intersection of circle which center is E and rectangle ABCD?
(1) 1
(2) 2
(3) 0
(4) 4

I have 3 questions:
a) Can I prove the answer with the geometry tool using axioms? Which are the axioms?
b) Can I prove it by another tool/s?
c) Can I prove the answer by proof by contradiction (reductoi ad impossibile)?

Happy day!
 
I would answer the question visually, by drawing some circles and seeing what happens. You can show that three of the cases (and another) can be obtained, and in doing so you can see informally why the fourth can't.

You are not expected to prove it formally, but you could, in various ways, by taking each of the cases that do work, and then considering the impossible case, which of course is the hardest to prove. One tool would be symmetry. You could also use reductio ad absurdum (if you know your Latin well enough). What axioms or theorems you use is up to you.

What have you tried?
 
shahar said:
View attachment 16680
The figure ABCD is a rectangle. E is the midpoint of the side AD.
Which answer from the following numbers cannot be the number of points of intersection of circle which center is E and rectangle ABCD?
(1) 1
(2) 2
(3) 0
(4) 4
Click to expand...
Notation \(\overline{AB}\) is a line segment whereas \(AB\) is its length.
If I were you, I would draw a rectangle( not a square), \{AB}>{AD}\), and get a compass.
Draw circles with radii: \(r_1=ED,~r_2=AB,~r_3=EB~\&~ r_4=DB.\)
 
pka said:
Notation \(\overline{AB}\) is a line segment whereas \(AB\) is its length.
If I were you, I would draw a rectangle( not a square), \{AB}>{AD}\), and get a compass.
Draw circles with radii: \(r_1=ED,~r_2=AB,~r_3=EB~\&~ r_4=DB.\)
Click to expand...
Sorry, but I didn't understand what do you mean or what do you want to show me?
Jomo said:
By symmetry there must be an even number of points of intersections unless the circle crosses the midpoint of BC. Do you see that?
What does this imply about your choices?
Click to expand...
I know the geometric picture that consists rectangle and circle is symmetry to the line that pass thorough E and perpendicular to AD so the even number of intersection points are even.
But also 1 (the wrong answer) and 3 can be the answer from this discretion.
I don't understand why the answer is 3 points of intersection and only one point.
 
Last edited:
Ignore the above post of mine.
Two question:
1. What is the symmetric tool and how it express in the problem?
2. What is the reasoning to do all the process of solving of pka?
 
shahar said:
Ignore the above post of mine.
Two question:
1. What is the symmetric tool and how it express in the problem?
2. What is the reasoning to do all the process of solving of pka?
Click to expand...
Did you "get a compass" and sketch as he had suggested? The reasoning will be self-evident as you go through/finish the process.
 
shahar said:
1. What is the symmetric tool and how it express in the problem?
Click to expand...
shahar said:
I know the geometric picture that consists rectangle and circle is symmetry to the line that pass thorough E and perpendicular to AD so the even number of intersection points are even.
Click to expand...
You answered your own question. You understand symmetry; the figure will be symmetrical, so that any points of intersection will be either on the line of symmetry, or paired. So the only odd numbers you can get will involve an intersection on that line (and possibly others). For each such possible point of intersection, what other intersections will there be?
 
What the purpose of circle with radius EB (r3) in the draw?
 
Maybe I didn't draw it correctly. O.K. So what the purpose, what does it give to me?
 
shahar said:
Maybe I didn't draw it correctly. O.K. So what the purpose, what does it give to me?
Click to expand...
for radius = r3

How many points of intersection do you have?

What are the sides of rectangle does the circle intersect?
 
O.K. Now I can see it!
I put the radii of 2 circles on the opposite side from the center incidentally.
Thank You for insist on checking my answer!
I think for the future telling to student that he need to check it is - Rule of Thumb.

It raise me the question in how many ways can a compass and ruler question will be understand wrong only by drawing radii in opposite direction. I know if I was a teacher I would notice my student to it.

Good Day!
 
shahar said:
What the purpose of circle with radius EB (r3) in the draw?
Click to expand...
First lets lable the midpoint of \(\overline{BC} \text{ with }F\)
If you draw the circles correctly then:
1) the circle with radius \(EF\) will intersect the rectangle in exactly three points.
2) the circle with radius \(EB\) will intersect the rectangle in exactly two points, \(B~\&~C\).
3) if \(EF<r_5<EB\) then the circle with radius \(r_5\) will intersect the rectangle in exactly four points.
 
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