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We are given that the figure \( ABCD \) is a square, and we are provided with the coordinates of points \( A(3, 5) \) and \( D(10, 5) \). The problem is to find the coordinates of point \( C \).

### Step-by-step analysis:

1. **Coordinates of \( A \) and \( D \):**
   - \( A(3, 5) \)
   - \( D(10, 5) \)

   These two points lie on a horizontal line, as they share the same \( y \)-coordinate, meaning the side \( AD \) is parallel to the \( x \)-axis.

2. **Length of side \( AD \):**
   Since \( ABCD \) is a square, the length of each side is the same. The distance between \( A \) and \( D \) (which is the side length of the square) is the difference in their \( x \)-coordinates:
   \[
   \text{Distance} = 10 - 3 = 7
   \]

3. **Coordinates of \( B \):**
   Point \( B \) is directly above point \( A \) since \( AB \) is vertical, and the distance between \( A \) and \( B \) is equal to the side length of the square (7). Therefore, the \( x \)-coordinate of \( B \) will be the same as \( A \)'s, and its \( y \)-coordinate will be:
   \[
   B(3, 5 + 7) = B(3, 12)
   \]

4. **Coordinates of \( C \):**
   Since \( BC \) is horizontal, the \( y \)-coordinate of \( C \) will be the same as that of \( B \), and its \( x \)-coordinate will be the same as \( D \)'s. Therefore, the coordinates of \( C \) will be:
   \[
   C(10, 12)
   \]

Thus, the coordinates of point \( C \) are \( \boxed{(10, 12)} \).

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Would you like more details on this solution or have any questions?

Here are five related questions to expand your understanding:

1. How would the coordinates change if \( ABCD \) was not a square but a rectangle?
2. What is the perimeter of the square \( ABCD \)?
3. How can you find the area of square \( ABCD \)?
4. What are the coordinates of the center of square \( ABCD \)?
5. How would you determine the slope of line \( AC \)?

**Tip:** When working with geometric figures on a coordinate plane, using the properties of shapes (like parallel sides and equal lengths) can simplify finding unknown coordinates.

What are the coordinates of point C in the square ABCD, given that A is at (3, 5) and D is at (10, 5)?

Learn how to find the coordinates of point C in the square ABCD, with given points A(3, 5) and D(10, 5). This step-by-step guide covers essential geometry concepts such as properties of squares and distance calculation, suitable for students in Grades 6-8.

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