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Jaycee is writing a coordinate proof to show that the diagonals of a rectangle bisect each other. She starts by assigning coordinates to a rectangle. Then she uses these coordinates to write the coordinates of the midpoint of each diagonal. She finds that the midpoints of the diagonals have the same coordinates, so the diagonals must bisect each other.

What are the coordinates of the midpoint of the diagonals of the rectangle?

Enter expressions in the box for the coordinates of the midpoint.

(, )

To solve this problem, we need to determine the coordinates of the midpoint of the diagonals of the rectangle. We are given a rectangle with vertices at:

- \( D(0, b) \)
- \( E(a, b) \)
- \( F(a, 0) \)
- \( G(0, 0) \)

The diagonals of the rectangle are:
1. From \( D(0, b) \) to \( F(a, 0) \)
2. From \( E(a, b) \) to \( G(0, 0) \)

### Step 1: Midpoint of Diagonal \( DF \)
The midpoint of a line segment with endpoints \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by the formula:
\[
\text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\]

For diagonal \( DF \), the endpoints are \( D(0, b) \) and \( F(a, 0) \). Applying the midpoint formula:

\[
\text{Midpoint of DF} = \left( \frac{0 + a}{2}, \frac{b + 0}{2} \right) = \left( \frac{a}{2}, \frac{b}{2} \right)
\]

### Step 2: Midpoint of Diagonal \( EG \)
Now, for diagonal \( EG \), the endpoints are \( E(a, b) \) and \( G(0, 0) \). Applying the midpoint formula again:

\[
\text{Midpoint of EG} = \left( \frac{a + 0}{2}, \frac{b + 0}{2} \right) = \left( \frac{a}{2}, \frac{b}{2} \right)
\]

### Conclusion:
Both diagonals have the same midpoint, which is \( \left( \frac{a}{2}, \frac{b}{2} \right) \). This confirms that the diagonals bisect each other.

The coordinates of the midpoint are \( \left( \frac{a}{2}, \frac{b}{2} \right) \).

Would you like any further details or clarifications?

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Here are five related questions:
1. How would you prove the diagonals of a square bisect each other?
2. What happens if the rectangle is not aligned with the axes?
3. How would the midpoint formula change if one of the vertices had negative coordinates?
4. How can you prove that the diagonals of a rhombus also bisect each other?
5. Can the midpoint formula be extended to three-dimensional geometry?

**Tip:** Always use the midpoint formula to check if two line segments bisect each other in any geometric shape!

Jaycee is writing a coordinate proof to show that the diagonals of a rectangle bisect each other. She starts by assigning coordinates to a rectangle. Then she uses these coordinates to write the coordinates of the midpoint of each diagonal. She finds that the midpoints of the diagonals have the same coordinates, so the diagonals must bisect each other. What are the coordinates of the midpoint of the diagonals of the rectangle?

Explore a coordinate proof demonstrating that the diagonals of a rectangle bisect each other. This problem applies the midpoint formula to show that both diagonals share the same midpoint, confirming that they bisect one another. Suitable for students in Grades 8-10.

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