ABCD is a trapezium in which AB is parallel to DC. Let E and F be the midpoints on AD and BC respectively. If EF = 10 cm and AB - DC = 4 cm, then what is the value of AB × DC?
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:
| Equation | Form |
|---|---|
| Equation 1 | \(AB + DC = 20\) |
| Equation 2 | \(AB - DC = 4\) |
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
| Concept | Description |
|---|---|
| Trapezium (Trapezoid) | A quadrilateral with at least one pair of parallel sides. The parallel sides are called bases. |
| Bases | The parallel sides of a trapezium (e.g., AB and DC in this problem). |
| Non-parallel sides | The other two sides of the trapezium (e.g., AD and BC). |
| Midpoint Segment (Median) | The line segment connecting the midpoints of the non-parallel sides. Its length is half the sum of the bases. |
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 \( \text{Area} = EF \times h \).
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