The diagram shows a rhombus ABCD. The points B and D have coordinates (2,10) and (6,2) respectively, and A lies on the x-axis. The mid-point of BD is M. Find, by calculation, the coordinates of each of M,A, and c.

To solve this problem, we need to determine the coordinates of points \( M \), \( A \), and \( C \) in the rhombus \( ABCD \).

### Step 1: Find the coordinates of the midpoint \( M \)
The midpoint \( M \) of the diagonal \( BD \) can be found using the midpoint formula:
\[
M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)
\]
Given the coordinates of \( B(2, 10) \) and \( D(6, 2) \), we substitute these values into the formula:
\[
M = \left(\frac{2 + 6}{2}, \frac{10 + 2}{2}\right) = \left(\frac{8}{2}, \frac{12}{2}\right) = (4, 6)
\]
So, the coordinates of \( M \) are \( (4, 6) \).

### Step 2: Find the coordinates of point \( A \)
Point \( A \) lies on the \( x \)-axis, so its \( y \)-coordinate is 0. Since \( M \) is the midpoint of \( AC \) (the other diagonal), we use the fact that \( A \) and \( C \) must be symmetric with respect to \( M \). This symmetry implies:
\[
x_A + x_C = 2 \times x_M \quad \text{and} \quad y_A + y_C = 2 \times y_M
\]

Let the coordinates of \( A \) be \( (x_A, 0) \), and those of \( C \) be \( (x_C, y_C) \). Since \( M(4, 6) \) is the midpoint, we have:
\[
x_A + x_C = 2 \times 4 = 8
\]
\[
0 + y_C = 2 \times 6 = 12 \quad \Rightarrow \quad y_C = 12
\]

### Step 3: Determine the coordinates of \( A \) and \( C \)
The line \( BD \) is a diagonal of the rhombus, and diagonals of a rhombus bisect each other at right angles. First, let's find the slope of \( BD \):
\[
\text{slope of } BD = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - 10}{6 - 2} = \frac{-8}{4} = -2
\]

Since the diagonals are perpendicular, the slope of \( AC \) will be the negative reciprocal of the slope of \( BD \):
\[
\text{slope of } AC = \frac{1}{2}
\]

Using the point-slope form for line \( AC \) with midpoint \( M(4, 6) \) and slope \( \frac{1}{2} \):
\[
y - 6 = \frac{1}{2}(x - 4)
\]
Simplify this equation:
\[
y = \frac{1}{2}x + 4
\]

Since \( A \) lies on the \( x \)-axis, \( y = 0 \) for point \( A \):
\[
0 = \frac{1}{2}x_A + 4
\]
Solve for \( x_A \):
\[
\frac{1}{2}x_A = -4 \quad \Rightarrow \quad x_A = -8
\]
So, the coordinates of \( A \) are \( (-8, 0) \).

### Step 4: Find the coordinates of \( C \)
Using the condition \( x_A + x_C = 8 \):
\[
-8 + x_C = 8 \quad \Rightarrow \quad x_C = 16
\]
And we already found \( y_C = 12 \). Therefore, the coordinates of \( C \) are \( (16, 12) \).

### Final Answer
- The coordinates of \( M \) are \( (4, 6) \).
- The coordinates of \( A \) are \( (-8, 0) \).
- The coordinates of \( C \) are \( (16, 12) \).

Would you like further explanation or have any questions?

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Here are five related questions:
1. How do you find the midpoint of a line segment given its endpoints?
2. What is the significance of the slopes of perpendicular lines?
3. How can the symmetry of a rhombus be used to find missing coordinates?
4. What properties of a rhombus help in determining its vertices?
5. How do you determine the equation of a line given a point and a slope?

**Tip:** Always check the relationship between the slopes of diagonals in quadrilaterals like rhombuses to ensure accuracy in calculations.

Learn how to find the coordinates of midpoint M, vertex A, and vertex C in rhombus ABCD. This problem involves using the midpoint formula, understanding rhombus properties, and applying the concept of perpendicular slopes. Suitable for students in Grades 9-12.

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