If there is a square with the points (2,9) and (9,2), what would the other two points be to fill in the whole first quadrant?

The points you've given are two opposite corners of the square. To find the other two points, we need to determine the direction and distance from one point to the other.

1. Calculate the difference in the x-coordinates (9-2=7) and the y-coordinates (2-9=-7). This tells us that we move 7 units to the right and 7 units down to get from (2,9) to (9,2).

2. To find the other two points, we need to make these same moves from each of our starting points.

3. If we start at (2,9) and move 7 units to the right, we end up at (9,9).

4. If we start at (9,2) and move 7 units up, we end up at (9,9) as well. So, (9,9) is one of our missing points.

5. If we start at (2,9) and move 7 units down, we end up at (2,2).

6. If we start at (9,2) and move 7 units to the left, we end up at (2,2) as well. So, (2,2) is the other missing point.

So, the four points that make up the square in the first quadrant are (2,9), (9,2), (9,9), and (2,2).

The points given are (2,9) and (9,2). These points form a diagonal of the square.

Step 1: Find the midpoint of the diagonal. The midpoint formula is ((x1+x2)/2, (y1+y2)/2). Substituting the given points, we get ((2+9)/2, (9+2)/2) = (5.5, 5.5).

Step 2: Find the length of the diagonal. The distance formula is sqrt((x2-x1)^2 + (y2-y1)^2). Substituting the given points, we get sqrt((9-2)^2 + (2-9)^2) = sqrt(49 + 49) = sqrt(98).

Step 3: The length of the side of the square is the diagonal divided by sqrt(2). So, the side length is sqrt(98)/sqrt(2) = sqrt(49) = 7.

Step 4: The other two points of the square are (x, y) coordinates where x and y are 5.5 plus or minus half the side length. Half the side length is 7/2 = 3.5.

So, the other two points are (5.5+3.5, 5.5-3.5) = (9, 2) and (5.5-3.5, 5.5+3.5) = (2, 9).

But these points are the same as the points given, so it seems there might be a mistake in the question. The points given already form a diagonal of the square, so the other two points should be different.

If we assume that the points given are two adjacent vertices of the square instead of the diagonal, then the other two points would be (2,2) and (9,9).