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To find the coordinates of point \( E \), which is the intersection point of the diagonals of the rectangle, we can use the property that the diagonals of a rectangle bisect each other. Therefore, point \( E \) will be the midpoint of the diagonals, which means it is the midpoint of both \( \overline{AC} \) and \( \overline{BD} \).

### Given coordinates:
- \( A(-3, 4) \)
- \( B(9, 10) \)
- \( D(1, -4) \)
- \( C(x, 2) \) — we do not know the value of \( x \), but can find it using midpoint properties.

The midpoint \( E \) of diagonal \( \overline{AC} \) and diagonal \( \overline{BD} \) should be the same. Let's first calculate the midpoint of \( \overline{BD} \).

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

#### Step 1: Midpoint of \( \overline{BD} \)
Using the coordinates \( B(9, 10) \) and \( D(1, -4) \), we calculate the midpoint \( E \) as follows:

\[
E = \left( \frac{9 + 1}{2}, \frac{10 + (-4)}{2} \right)
= \left( \frac{10}{2}, \frac{6}{2} \right)
= (5, 3)
\]

#### Step 2: Midpoint of \( \overline{AC} \)
Now, we know that point \( E(5, 3) \) is also the midpoint of \( \overline{AC} \). Using the coordinates \( A(-3, 4) \) and \( C(x, 2) \), we can set up the midpoint equation:

\[
E = \left( \frac{-3 + x}{2}, \frac{4 + 2}{2} \right) = (5, 3)
\]

From the second coordinate equation:
\[
\frac{4 + 2}{2} = 3 \quad \text{(This is true, so no need to solve further for the y-coordinate)}
\]

Now, solving for the x-coordinate:
\[
\frac{-3 + x}{2} = 5
\]

Multiplying both sides by 2:
\[
-3 + x = 10
\]

Solving for \( x \):
\[
x = 13
\]

Thus, the coordinates of point \( C \) are \( C(13, 2) \).

### Final Answer:
The coordinates of point \( E \), where the diagonals intersect, are \( \boxed{(5, 3)} \).

Would you like further details on this solution, or have any other questions?

Here are some related questions to explore:
1. How can we verify that the diagonals bisect each other in a rectangle?
2. What other properties of rectangles are important in coordinate geometry?
3. How can you find the equation of a diagonal using two points?
4. How do you calculate the area of a rectangle given the vertices?
5. What is the significance of midpoints in different types of quadrilaterals?

**Tip:** Remember that in any parallelogram, including rectangles, the diagonals always bisect each other.

Find the coordinate of point E where the diagonals of the rectangle intersect.

Learn how to find the intersection point of the diagonals of a rectangle by using the midpoint formula. This problem explores key concepts in coordinate geometry and rectangle properties, suitable for students in Grades 8-10.

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