Thank you for using the timer!
We noticed you are actually not timing your practice. Click the START button first next time you use the timer.
There are many benefits to timing your practice, including:
A certain square is to be drawn on a coordinate plane. One [#permalink]
27 Mar 2007, 18:54
This topic is locked. If you want to discuss this question please re-post it in the respective forum.
A certain square is to be drawn on a coordinate plane. One of the vertices must be on the origin, and the square is to have an area of 100. If all coordinates of the vertices must be integers, how many different ways can this square be drawn?
1. Set1 - 4 squares, one in each quadrant. Vertices- (0,0) (0,10) (0,-10) (10,0) (-10,0)
2 Set2 - Rotate the set 1 of the squares in clockwise so that the the vertex of (10,0) becomes (8,6). There will be 4 squares with this orientation.
3 Set3 -Rotate the set1 squares further so that the the vertex of (10,0) becomes (6,8).There will be 4 squares with this orientation.
Hence there will be 4+4+4 = 12 squares.
It's hard to imagine that way without putting it diagrammatically.
Attached is the diagram. It;s not accurate but will certainly help you visualizing the squares orientation.
Re: how many squares in a plane [#permalink]
28 Mar 2007, 14:55
A certain square is to be drawn on a coordinate plane. One of the vertices must be on the origin, and the square is to have an area of 100. If all coordinates of the vertices must be integers, how many different ways can this square be drawn? 4 6 8 10 12
Such a square must have a point P(x,y) that is sqrt(200) from the origin
i.e. x^2+y^2=200 where x and y are integers
|y| could be 2,10,14
So, I guess there will be 12 such squares
Let's look at the squares in an organized way. Let P(x,y) be the vertex of the side OP we would encounter if we started at the y axis and looked clockwise. How many such pairs (x,y) are there? |x|=sqrt(100-y^2), where x and y are both integers, so we can make a chart:
Fig's explanation is good. There is also a visual way to do this:
First, if one vertex is on integer coordinates, then all of them will be. You can draw a picture and use geometric similarities to see this. Thus, the problem reduces to rotating a vertex adjacent to the origin in a circle, and counting the times it passes through an integer coordinate. This then, will give (going counterclockwise) (10,0), (8,6), (6,8), (0,10), (-6,8), ... (8,-6) for a total of 12.