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.