Trapezium Side Lengths Calculation

We are given the area ($A$), height ($h$), and the difference between the parallel sides of a trapezium. We need to find the lengths of these parallel sides.

Let the lengths of the parallel sides be $a$ and $b$ (in cm), where $a > b$. Let the height be $h$ (in cm) and the area be $A$ (in cm\(^2\)).

Given Information:

• Difference between parallel sides: $a - b = 9$ cm
• Height: $h = 52$ cm
• Area: $A = 988$ cm\(^2\)

Area Formula and Calculation:

The formula for the area of a trapezium is:

$ A = \frac{1}{2}(a+b)h $

Substitute the given values into the formula:

$ 988 = \frac{1}{2}(a+b)(52) $

Simplify the equation:

$ 988 = (a+b)(26) $

Solve for the sum of the parallel sides ($a+b$):

$ a+b = \frac{988}{26} $

$ a+b = 38 $

Solving for Parallel Sides:

Now we have a system of two linear equations:

1. $a + b = 38$
2. $a - b = 9$

Add the two equations together to eliminate $b$:

$ (a+b) + (a-b) = 38 + 9 $

$ 2a = 47 $

Solve for $a$:

$ a = \frac{47}{2} $

$ a = 23.5 \text{ cm} $

Substitute the value of $a$ back into the first equation ($a+b=38$):

$ 23.5 + b = 38 $

Solve for $b$:

$ b = 38 - 23.5 $

$ b = 14.5 \text{ cm} $

Conclusion:

The lengths of the parallel sides are 23.5 cm and 14.5 cm.