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The problem involves a rectangle \( ABCD \) with diagonals that intersect at point \( O \). Given that \( DO = 4x + 8 \) and \( CO = 15x - 3 \), and since diagonals of a rectangle bisect each other, it means:

\[
DO = CO
\]

Thus, we can set the expressions for \( DO \) and \( CO \) equal to each other:

\[
4x + 8 = 15x - 3
\]

Now, solve for \( x \):

1. Subtract \( 4x \) from both sides:

\[
8 = 11x - 3
\]

2. Add 3 to both sides:

\[
11 = 11x
\]

3. Divide by 11:

\[
x = 1
\]

Thus, the value of \( x \) is \( 1 \).

Would you like more details or have any further questions? Here are five related questions to expand on this:

1. How can you check if the diagonals of any parallelogram bisect each other?
2. If \( x = 1 \), what are the lengths of \( DO \) and \( CO \)?
3. How would the solution change if the diagonals intersected at an angle instead of bisecting each other?
4. How can the properties of a rectangle help in solving other geometric problems involving diagonals?
5. Can the same approach be used for other quadrilaterals, such as a rhombus or trapezoid?

**Tip:** Remember that in all parallelograms, diagonals bisect each other, and in rectangles, the diagonals are equal. This property can simplify many geometry problems!

Given the rectangle ABCD as shown, find the value for x if DO = 4x + 8 and CO = 15x - 3.

Solve for x in a rectangle where diagonals bisect each other. The problem involves algebra and geometry. Given DO = 4x + 8 and CO = 15x - 3, this step-by-step solution demonstrates how to solve the equation by equating the diagonals.

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