Solving Trapezium Problems with Midpoints

Let's break down this geometry problem involving a trapezium and its midpoints. We are given a trapezium ABCD where side AB is parallel to side DC. E and F are the midpoints of the non-parallel sides AD and BC, respectively.

We are provided with two key pieces of information:

• The length of the line segment EF is 10 cm.
• The difference between the lengths of the parallel sides AB and DC is 4 cm (AB - DC = 4 cm).

Our goal is to find the value of the product of the lengths of the parallel sides, i.e., AB × DC.

Midpoint Segment Property of a Trapezium

A crucial property of a trapezium states that the line segment connecting the midpoints of the non-parallel sides is parallel to the parallel sides and its length is equal to half the sum of the lengths of the parallel sides. Mathematically, this can be written as:

\(EF = \frac{1}{2}(AB + DC)\)

Applying the Given Information

We can substitute the given length of EF into this formula:

\(10 = \frac{1}{2}(AB + DC)\)

Multiplying both sides by 2, we get:

\(20 = AB + DC \quad \text{(Equation 1)}\)

We are also given the difference between the lengths of the parallel sides:

\(AB - DC = 4 \quad \text{(Equation 2)}\)

Solving the System of Equations

Now we have a system of two linear equations with two variables, AB and DC:

We can solve this system by adding the two equations. Adding Equation 1 and Equation 2 eliminates DC:

\((AB + DC) + (AB - DC) = 20 + 4\)

\(AB + AB + DC - DC = 24\)

\(2AB = 24\)

Dividing both sides by 2:

\(AB = \frac{24}{2}\)

\(AB = 12 \text{ cm}\)

Now substitute the value of AB (12 cm) into either Equation 1 or Equation 2 to find DC. Using Equation 1:

\(12 + DC = 20\)

Subtract 12 from both sides:

\(DC = 20 - 12\)

\(DC = 8 \text{ cm}\)

Calculating the Product AB × DC

We have found the lengths of the parallel sides:

• AB = 12 cm
• DC = 8 cm

We need to find the value of AB × DC:

\(AB \times DC = 12 \times 8\)

\(AB \times DC = 96\)

The value of AB × DC is 96. The unit provided in the options is square cm, which typically denotes area, but here it represents the numerical product of the lengths.

Revision Table: Trapezium Key Concepts

Additional Information on Trapezium Properties

Beyond the midpoint segment property, trapeziums have other interesting characteristics:

• The sum of consecutive interior angles between a parallel side and a non-parallel side is 180°. For example, \( \angle DAB + \angle ADC = 180^\circ \) and \( \angle ABC + \angle BCD = 180^\circ \).
• The diagonals of a trapezium intersect each other. If the trapezium is isosceles (non-parallel sides are equal), the diagonals are equal in length and intersect such that the segments formed are proportional.
• The area of a trapezium is given by the formula: \( \text{Area} = \frac{1}{2} \times (\text{sum of parallel sides}) \times (\text{height}) \). In this problem, \( \text{Area} = \frac{1}{2}(AB + DC) \times h \), where \(h\) is the height. Note that \( EF = \frac{1}{2}(AB + DC) \), so the area can also be expressed as \( \text{Area} = EF \times h \).