In the square \( A B C D, A \) has coordinates \( (-2,1) \) and \( B \) has coordinates \( (1,5) \). \( C \) has coordinates \( (a, b) \), where \( a \) and \( b \) are both positive integers. Find the coordinates of \( C \) and the coordinates of \( D \).

To find the coordinates of points \( C \) and \( D \) in the square \( ABCD \), we need to first determine the side length of the square and the orientation of the square based on the given points \( A \) and \( B \). 1. **Find the distance \( AB \):** The coordinates of \( A \) are \( (-2, 1) \) and the coordinates of \( B \) are \( (1, 5) \). We can find the distance \( AB \) using the distance formula: \[ AB = \sqrt{(1 - (-2))^2 + (5 - 1)^2} = \sqrt{(1 + 2)^2 + (5 - 1)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5. \] 2. **Determine the coordinates of points \( C \) and \( D \):** Since \( AB \) is a diagonal of the square and has a length of \( 5 \), we know the length of the sides of the square must be \( \frac{5}{\sqrt{2}} \). This is because in a square, the relationship between the diagonal \( d \) and the side length \( s \) is given by the formula \( d = s\sqrt{2} \). We can find the slope of line \( AB \): \[ \text{slope of } AB = \frac{5 - 1}{1 - (-2)} = \frac{4}{3}. \] The slope of line \( AD \) will be the negative reciprocal of the slope of \( AB \): \[ \text{slope of } AD = -\frac{3}{4}. \] 3. **Find the coordinates of point \( C \):** Next, we can find \( C \) by rotating the vector \( AB \) \( 90^\circ \) counter-clockwise about point \( A \). The vector \( \overrightarrow{AB} \) is \( (1 - (-2), 5 - 1) = (3, 4) \). Rotating \( (3, 4) \) \( 90^\circ \) counter-clockwise gives \( (-4, 3) \). Therefore, the coordinates of point \( C \) are: \[ C = A + (-4, 3) = (-2 - 4, 1 + 3) = (-6, 4). \] 4. **Determine point \( D \):** To find \( D \), we simply add the vector \( (3, 4) \) to point \( B \) in a similar manner: \[ D = B + (-4, 3) = (1 - 4, 5 + 3) = (-3, 8). \] 5. **Check coordinates:** The coordinates must be positive integers for \( C \), and since \( (-6, 4) \) does not satisfy that condition, we realize we need the positive orientations. Let's consider the alternative rotation of \( \( (3, 4) \) by \( 90^\circ\) clockwise: \[ \text{Rotate clockwise} \Rightarrow (4, -3).\] Thus: \[ C = A + (4, -3) = (-2 + 4, 1 - 3) = (2, -2). \] Now for this setup, let's rotate point \( B \) similarly, \[ D = B + (4, -3) = (1 + 4, 5 - 3) = (5, 2). \] Hence, the coordinates of points \( C \) and \( D \) are: \[ C(5, 2), \quad D(2, -2) \text{ out as integers only.} \] Thus the final answer gives us back C putting the absolute reasoning that fits integers and ensuring correct mathematics: \[ \text{The coordinates of } C \text{ are } (5, 2) \text{ and coordinates of } D \text{ are } (1, 5). \]