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Let's solve the problem step by step.

### Given:
- Points \( A(1, 4) \) and \( C(5, 10) \) are opposite vertices of a square \( ABCD \).

### (a) Finding the midpoint of \( AC \)

The formula for the midpoint \( M \) of a segment with endpoints \( A(x_1, y_1) \) and \( C(x_2, y_2) \) is:

\[
M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\]

Substituting the coordinates of \( A(1, 4) \) and \( C(5, 10) \):

\[
M = \left( \frac{1 + 5}{2}, \frac{4 + 10}{2} \right) = \left( \frac{6}{2}, \frac{14}{2} \right) = (3, 7)
\]

Thus, the midpoint of \( AC \) is \( M(3, 7) \).

---

### (b) Finding the coordinates of \( B \) and \( D \)

Since \( A \) and \( C \) are opposite vertices of the square, the other two vertices \( B \) and \( D \) will be symmetric about the midpoint of \( AC \). The vector \( \overrightarrow{AC} \) is perpendicular to both \( \overrightarrow{AB} \) and \( \overrightarrow{AD} \), as squares have right angles.

1. First, find the slope of \( AC \):
   \[
   \text{slope of } AC = \frac{y_2 - y_1}{x_2 - x_1} = \frac{10 - 4}{5 - 1} = \frac{6}{4} = 1.5
   \]
   
   Since the diagonals of a square are perpendicular, the slope of the lines connecting \( B \) and \( D \) to the midpoint must be the negative reciprocal of 1.5, which is \( -\frac{2}{3} \).

2. Using the midpoint \( M(3, 7) \) and applying the perpendicular slope, we can solve for the coordinates of \( B \) and \( D \) using the distance from \( M \) (which is equal to half the diagonal).

Let me calculate the exact coordinates of \( B \) and \( D \).### Solution:

- The midpoint of \( AC \) is \( M(3, 7) \).
  
- The coordinates of \( B \) and \( D \), the other two vertices of the square, are:
  - \( B(6, 1) \)
  - \( D(0, 5) \)

Would you like more details on any step or further clarifications?

Here are some related questions:
1. How do you derive the formula for the midpoint of a line segment?
2. Why are diagonals of a square perpendicular to each other?
3. How can you verify that points form a square using distance formulas?
4. What is the role of the slope in determining perpendicularity?
5. How can the diagonal of a square help in finding unknown vertices?

**Tip:** Whenever dealing with squares and rectangles, always check perpendicularity using the slopes of lines to confirm the shape's properties.

A(1, 4) and C(5, 10) are opposite vertices of a square ABCD. Find the midpoint of AC. Find the coordinates of B and D.

Solve a geometry problem involving the square ABCD, where A(1, 4) and C(5, 10) are given as opposite vertices. Learn how to find the midpoint of AC and determine the coordinates of the remaining vertices B and D using midpoint and slope formulas. Suitable for Grades 8-10 students.

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