The problem shows a rectangle $ABCD$ with diagonals $AC$ and $BD$ intersecting at a point. For a rectangle, the diagonals are equal and bisect each other. Let's analyze the problem step-by-step:

1. Property of the rectangle: In a rectangle, diagonals are equal, i.e., $AC = BD$.

2. The diagonals are given as:

   • $AC = 3x + 2$
   • $BD = 5x - 6$

3. Set the diagonals equal to each other:

$3x + 2 = 5x - 6$

4. Solving the equation:

$3x + 2 = 5x - 6$ $2 + 6 = 5x - 3x$ $8 = 2x$ $x = 4$

So, the value of $x$ is:

$x = 4$

Let me know if you would like a more detailed explanation or have any questions!

Related Questions:

1. What is the length of diagonal $AC$ for $x = 4$?
2. What is the length of diagonal $BD$ for $x = 4$?
3. What other properties do diagonals of rectangles possess?
4. How can we verify that the diagonals bisect each other in this figure?
5. Can the same property of diagonals be applied to a square?

Tip:

In any parallelogram (including rectangles), the diagonals bisect each other, which means the point of intersection divides each diagonal into two equal halves.