Two points of a rhombus ABCD are A= (2, 7) and C = (4, 3). Find the coordinates of B and D if the x-coordinate of B is -3

To find the coordinates of points \( B \) and \( D \) of the rhombus \( ABCD \) given points \( A = (2, 7) \) and \( C = (4, 3) \), and that the \( x \)-coordinate of \( B \) is \(-3\), we can use the properties of a rhombus.

### Step 1: Find the midpoint \( O \) of the diagonal \( AC \)
The midpoint \( O \) of diagonal \( AC \) can be found using the midpoint formula:
\[
O = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)
\]
Substituting the coordinates of \( A = (2, 7) \) and \( C = (4, 3) \):
\[
O = \left(\frac{2 + 4}{2}, \frac{7 + 3}{2}\right) = \left(3, 5\right)
\]

### Step 2: Use the midpoint to find the equations for \( B \) and \( D \)
Since \( O \) is also the midpoint of the diagonal \( BD \), the coordinates of \( B \) and \( D \) satisfy:
\[
\left(\frac{x_B + x_D}{2}, \frac{y_B + y_D}{2}\right) = (3, 5)
\]
This gives us two equations:
\[
\frac{x_B + x_D}{2} = 3 \quad \text{and} \quad \frac{y_B + y_D}{2} = 5
\]
Multiplying both equations by 2:
\[
x_B + x_D = 6 \quad \text{and} \quad y_B + y_D = 10
\]

### Step 3: Use the given \( x \)-coordinate of \( B \)
We know that \( x_B = -3 \), so substitute it into the equation for \( x_B + x_D \):
\[
-3 + x_D = 6 \implies x_D = 9
\]
So, the coordinates of \( B \) and \( D \) are:
\[
x_B = -3 \quad \text{and} \quad x_D = 9
\]

### Step 4: Determine the \( y \)-coordinates for \( B \) and \( D \)
Substitute \( y_B + y_D = 10 \) into the equation. Let \( y_B = y \), then \( y_D = 10 - y \).

To maintain the property of a rhombus (equal-length sides), use the distance formula for \( AB \) and \( BC \).

### Step 5: Calculate \( y_B \) and \( y_D \)
Since the diagonals of a rhombus bisect each other at right angles, we can use the perpendicular slope relationship to find \( y \).

Let the slope of \( AC \) be \( m_{AC} \) and the slope of \( BD \) be \( m_{BD} \). The slopes should satisfy:
\[
m_{AC} \times m_{BD} = -1
\]
Calculate \( m_{AC} \):
\[
m_{AC} = \frac{3 - 7}{4 - 2} = \frac{-4}{2} = -2
\]
Then \( m_{BD} \) should be:
\[
m_{BD} = \frac{y_D - y_B}{x_D - x_B} = \frac{10 - 2y}{12} = \frac{5 - y}{6}
\]
Set up the equation:
\[
-2 \times \frac{5 - y}{6} = -1
\]
Simplifying:
\[
\frac{5 - y}{3} = 1 \implies 5 - y = 3 \implies y = 2
\]
So, \( y_B = 2 \) and \( y_D = 10 - 2 = 8 \).

### Final Coordinates:
- \( B = (-3, 2) \)
- \( D = (9, 8) \)

Would you like any further explanation, or do you have any questions?

Here are five questions for practice:
1. What would be the coordinates of \( B \) if \( x_B \) were given as \( 3 \) instead?
2. How would the process change if the rhombus were rotated by 45 degrees around the origin?
3. What are the properties of a rhombus that distinguish it from other quadrilaterals?
4. How would you find the area of rhombus \( ABCD \) given these coordinates?
5. What if the midpoint of diagonal \( AC \) was not known? How could you determine the coordinates of \( B \) and \( D \)?

**Tip:** Always double-check your work by verifying that the properties of the geometric figure (like equal sides in a rhombus) hold true with your calculated points.

Two points of a rhombus ABCD are A = (2, 7) and C = (4, 3). Find the coordinates of B and D if the x-coordinate of B is -3.

Learn how to find the coordinates of points B and D of a rhombus ABCD, given points A = (2, 7) and C = (4, 3), and that the x-coordinate of B is -3. This solution uses midpoint, distance, and slope formulas while applying the properties of a rhombus. Suitable for Grades 9-11 geometry students.

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