In a trapezium ABCD, AB ‖ CD. If OA = x + 5, OB = x – 7, OC = x + 3 and OD =
x – 2, then find x

To solve for \( x \) in this trapezium problem, we will use the fact that in any trapezium where \( AB \parallel CD \), if the diagonals are drawn and intersect at point \( O \), the following property holds:

\[
\frac{OA}{OB} = \frac{OC}{OD}
\]

Given:
- \( OA = x + 5 \)
- \( OB = x - 7 \)
- \( OC = x + 3 \)
- \( OD = x - 2 \)

Using the property, we can set up the equation:

\[
\frac{OA}{OB} = \frac{OC}{OD}
\]

Substituting the given values:

\[
\frac{x + 5}{x - 7} = \frac{x + 3}{x - 2}
\]

Now, we will solve this equation step-by-step.

1. Cross-multiply:

\[
(x + 5)(x - 2) = (x - 7)(x + 3)
\]

2. Expand both sides:

\[
(x + 5)(x - 2) = x^2 - 2x + 5x - 10 = x^2 + 3x - 10
\]
\[
(x - 7)(x + 3) = x^2 + 3x - 7x - 21 = x^2 - 4x - 21
\]

3. Set the expanded forms equal to each other:

\[
x^2 + 3x - 10 = x^2 - 4x - 21
\]

4. Simplify by subtracting \( x^2 \) from both sides:

\[
3x - 10 = -4x - 21
\]

5. Add \( 4x \) to both sides:

\[
7x - 10 = -21
\]

6. Add 10 to both sides:

\[
7x = -11
\]

7. Divide by 7:

\[
x = -\frac{11}{7}
\]

Thus, the value of \( x \) is \( -\frac{11}{7} \).

Do you want more details, or do you have any questions?

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Here are 5 related questions:
1. What is the formula for the area of a trapezium?
2. How do the properties of parallel sides in a trapezium affect its diagonals?
3. Can the diagonals of a trapezium be equal? Under what conditions?
4. What are the different types of trapeziums, and how do their properties differ?
5. How do you derive the equation of a line parallel to another in geometry?

**Tip:** In a trapezium, the diagonals generally do not bisect each other, but they divide each other proportionally when the sides are parallel.

In a trapezium ABCD, AB ‖ CD. If OA = x + 5, OB = x – 7, OC = x + 3 and OD = x – 2, then find x.

Learn how to solve for x in a trapezium ABCD where AB is parallel to CD, and the diagonals divide each other proportionally. This problem uses algebraic techniques to set up and solve the proportionality equation. Suitable for students in Grades 9-11.

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