So i have question about we use the x^2 + Y^2 = 10^2 concept to solve this problem. These other values that you mentioned 6^2 + 8^2 may solve to 100 but isn't Area of a square = l*w. In this case, 6*8 isn't 100 so how are we justifying this? I've looked at the drawn out images as well, and I know I'm missing something but don't know what it is. Can you please clarify? Thanks!
jpr200012 wrote:
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?
(A) 4
(B) 6
(C) 8
(D) 10
(E) 12
This question becomes much easier if you visualize/draw it.
Let the origin be O and one of the vertices be A. Now, we are told that length of OA must be 10 (area to be 100). So if the coordinates of A is (x, y) then we would have \(x^2+y^2=100\) (distance from the origin to the point A(x, y) can be found by the formula \(d^2=x^2+y^2\))
Now, \(x^2+y^2=100\) has several integer solutions for \(x\) and \(y\), so several positions of vertex A, note that when vertex A has integer coordinates other vertices also have integer coordinates. For example imagine the case when square rests on X-axis to the right of Y-axis, then the vertices are: A(10,0), (10,10), (0,10) and (0,0).
Also you can notice that 100=6^2+8^2 and 100=0^2+10^2, so \(x\) can tale 7 values: -10, -8, -6, 0, 6, 8, 10. For \(x=-10\) and \(x=10\), \(y\) can take only 1 value 0, but for other values of \(x\), \(y\) can take two values positive or negative. For example: when \(x=6\) then \(y=8\) or \(y=-8\). This gives us 1+1+5*2=12 coordinates of point A:
\(x=10\) and \(y=0\), imagine this one to be the square which rests on X-axis and to get the other options rotate OA anticlockwise to get all possible cases;
\(x=8\) and \(y=6\);
\(x=6\) and \(y=8\);
\(x=0\) and \(y=10\);
\(x=-6\) and \(y=8\);
\(x=-8\) and \(y=6\);
\(x=-10\) and \(y=0\);
\(x=-8\) and \(y=-6\);
\(x=-6\) and \(y=-8\);
\(x=0\) and \(y=-10\);
\(x=6\) and \(y=-8\);
\(x=8\) and \(y=-6\).
Answer: E.