Determine if point lies on inscribed square

In summary, the conversation discusses how to determine if a point lies on the sides of a square inscribed on a circle with only the radius of the circle and the circle being centered at the origin given. The idea of finding the equation for each side and testing the given point is suggested, but it is realized that there can be multiple ways to inscribe a square. The conversation then delves into finding the radius of the circle and how points inside the circle do not belong to an inscribed square, while points in between the circles do belong to an inscribed square. Using a free program called geogebra is recommended to better understand the concept. Ultimately, it is explained that rotating the square creates an inscribed circle, with the sides
  • #1
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How do you determine if a point lies on the sides of a square inscribed on a circle ?
I'm given only the radius of the circle and the circle is centered at the origin.
My idea was to find the equation for each side and then to test the given point.
However, I just realized that you can inscribe a square in many different ways.
I'm stumped on how to do this.
 
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  • #2
Do you mean "determine if a point lies on A certain square" or on a specific square? If you need to prove the last, then you really need some more information.

The first question is a bit more fun. Try to see it intuitively first. Draw an inscribed square and mentally change the square to another inscribed square. You will see that you get a second (smaller) circle. Points lying inside the circle do not belong to a inscribed square, and points lying inbetween the cricles do belong to a insribed square.
The only thing you need to do now is to find the radius of the circle.

I'm sorry if you understood nothing of what I tried to describe, but it's hard to explain. I suggest download the free program geogebra and play a bit with it. You'll see very soon what I mean...
 
  • #3
Its the first question.
I do not understand how you can get a smaller circle when you shift the square. Maybe I'm doing it wrong but I thought that when you shift the square, the end points of the square move along the circle. The diagonals of the square will be equal to the diameter of the circle and will remain the same regardless of the square is drawn.
 
  • #4
You don't shift the square, but you rotate the square. But perhaps you meant that.
It's true that the endpoints of the square will still lie on the circle, but the sides of the square will create an inscribed circle. The side of every inscribed square will be a tangent line to this inscribed circle. This fact would make it easy to find an equation for this circle...
 
  • #5
Ok, I get it now. I seem to vaguely remember an animation that showed this.
 

1. How do you determine if a point lies on an inscribed square?

To determine if a point lies on an inscribed square, you need to first locate the center of the square. Then, calculate the distance between the center and the given point. If the distance is equal to the distance between the center and any of the square's vertices, then the point lies on the inscribed square.

2. What is an inscribed square?

An inscribed square is a square that is drawn inside another shape, such as a circle or a triangle, in a way that all four of its vertices touch the edges of the larger shape.

3. What is the formula for calculating the distance between two points?

The formula for calculating the distance between two points is the Pythagorean theorem, which states that the square of the distance between two points is equal to the sum of the squares of the differences between their coordinates.

4. Can a point lie on an inscribed square if it is not exactly on one of its vertices?

No, a point can only lie on an inscribed square if it is exactly on one of its vertices. If the point is slightly off, it will not be considered as lying on the square.

5. What are some real-life applications of determining if a point lies on an inscribed square?

Determining if a point lies on an inscribed square has various real-life applications, such as in geometry and construction. It can also be used in navigation and map-making to accurately plot points and determine distances between them. Additionally, this concept is important in computer graphics and design when creating 2D and 3D shapes and models.

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