RENT is a rectangle. Its diagonals meets at O. Find x if `OR=2x+3` and `OT=3x+2`.

Answer

To solve the problem, we need to find the value of \( x \) given that \( OR = 2x + 3 \) and \( OT = 3x + 2 \) in rectangle RENT where the diagonals meet at point O.

### Step-by-step Solution:

1. **Understand the properties of a rectangle**: In a rectangle, the diagonals bisect each other. This means that the lengths of the segments from the intersection point O to the vertices of the rectangle are equal.

2. **Set up the equation**: Since \( OR \) and \( OT \) are segments of the diagonals that bisect each other, we can set them equal to each other:
\[ OR = OT \]

3. **Substitute the expressions for \( OR \) and \( OT \)**:
\[ 2x + 3 = 3x + 2 \]

4. **Rearrange the equation**: To solve for \( x \), we first move all terms involving \( x \) to one side and constant terms to the other side:
\[ 2x + 3 - 2 = 3x \]
This simplifies to:
\[ 2x + 1 = 3x \]

5. **Isolate \( x \)**: Now, subtract \( 2x \) from both sides:
\[ 1 = 3x - 2x \]
This simplifies to:
\[ 1 = x \]

6. **Conclusion**: Therefore, the value of \( x \) is:
\[ x = 1 \]