Understanding the Problem
In trapezium ABCD, with AB parallel to DC, we have the diagonals AC and BD intersecting at point O. The segments are defined as follows:
- OD = x - 2
- OC = x + 3
- OA = x + 5
- OB = x - 1
We need to find the value of x.
Applying the Properties of Trapezium
In a trapezium, the diagonals divide each other proportionally, which gives us the following relationship:
- \( \frac{OA}{OC} = \frac{OB}{OD} \)
Setting Up the Equation
Substituting the values into the proportion gives:
- \( \frac{x+5}{x+3} = \frac{x-1}{x-2} \)
Cross-Multiplying
Now, we will cross-multiply to solve for x:
- \( (x + 5)(x - 2) = (x + 3)(x - 1) \)
Expanding Both Sides
Expanding the left side:
- \( x^2 - 2x + 5x - 10 = x^2 + 3x - x - 3 \)
This simplifies to:
- \( x^2 + 3x - 10 = x^2 + 2x - 3 \)
Solving the Equation
Now, we simplify further:
- \( 3x - 10 = 2x - 3 \)
Subtracting \( 2x \) from both sides gives:
- \( x - 10 = -3 \)
Adding 10 to both sides yields:
- \( x = 7 \)
Conclusion
The value of x is 7. The diagonals of the trapezium are proportionally divided, confirming the properties of trapezium ABCD.