Coordinate system

1. Aug 3, 2010

thereddevils

1. The problem statement, all variables and given/known data

I will to explain this without a diagram. Consider a coordinate system(1st quadrant) where the x and y-axis both stop at 7 units (i mean the boundary), there are two points P(3,3) and Q(4,4). How many points can R be positioned such that PQR is an isosceles triangle.

2. Relevant equations

3. The attempt at a solution

I got 6.

2. Aug 3, 2010

Mentallic

Well I had quite a few and started counting them... Eventually I counted up to infinite different possible points :tongue:

There are actually 3 paths the point R can trace, a line and 2 circles. See if you can find them.

3. Aug 3, 2010

HallsofIvy

Staff Emeritus
Then you haven't given all the information. Are you talking about a "mesh" where the points must have integer components? As you give it, any point on the perpendicular bisector of PQ works and there are an infinite number of them.

4. Aug 3, 2010

thereddevils

yeah sorry, it must be of integer components where you can precisely represent that particular point on the plane.

My 6 points are

(5,2) , (2,5) , (6,1) , (1,6) , (7,0) , (0,7)

5. Aug 3, 2010

Mentallic

Yet the distance between P and Q is not an integer.
Anyway, this means we have the two equations

$$\sqrt{(x-4)^2+(y-4)^2}=n$$

$$\sqrt{(x-3)^2+(y-3)^2}=n$$

where n is an integer, to solve simultaneously.
After simplifying, we find the perpendicular bisector of PQ to be the line x+y-7=0
Now just use all possible integer values of x and y such that x+y=7.

I don't know if I should scrap the idea of the point R can trace that are circles rather than this line, because that means we have to use a non-integer value for another length.

6. Aug 4, 2010

thereddevils

sigh.. 8 points then including (4,3) , (3,4)