In this question, we are given that:

• $AC = 5x - 24$
• $AO = 3$
• $O$ is the midpoint of diagonal $AC$ because the diagonals of a rectangle bisect each other.

This means: $AO = \frac{AC}{2}.$

Let's proceed to solve for $x$.

Step-by-step solution:

1. Since $AO$ is half of $AC$, we write: $AO = \frac{AC}{2}.$

2. Substitute the known values of $AO$ and $AC$: $3 = \frac{5x - 24}{2}.$

3. Multiply through by 2 to eliminate the fraction: $6 = 5x - 24.$

4. Solve for $x$:

   • Add 24 to both sides: $6 + 24 = 5x.$ $30 = 5x.$

   • Divide both sides by 5: $x = 6.$

Final Answer:

$x = 6.$

Would you like me to verify this solution or explain any step further? Let me know!

Related Questions:

1. What are the properties of diagonals in a rectangle?
2. How do you find the midpoint of a line segment using its coordinates?
3. If $BD$ were given, how would you determine $BO$ or $DO$?
4. How would the problem change if $ABCD$ were a rhombus instead of a rectangle?
5. If $AC$ is increased to $5x - 10$, how would $x$ change?

Tip: In rectangles, diagonals are equal in length and bisect each other. This property is useful for solving midpoint and diagonal-related problems.