The area of a trapezium is 18 sq.cm. Its height and base are 3 cm and 5 cm respectively. Find the length of the side parallel to the base.
7 cm
Understanding the Trapezium and its Area
A trapezium (also known as a trapezoid in some regions) is a quadrilateral with at least one pair of parallel sides. These parallel sides are often referred to as the bases of the trapezium, and the perpendicular distance between them is called the height or altitude.
The area of a trapezium is calculated using a specific formula that relates its parallel sides and its height.
Formula for the Area of a Trapezium
The formula to find the area of a trapezium is:
\( \text{Area} = \frac{1}{2} \times (\text{Sum of parallel sides}) \times \text{Height} \)
If we denote the lengths of the two parallel sides as \(b_1\) and \(b_2\), and the height as \(h\), the formula can be written as:
\( \text{Area} = \frac{1}{2} \times (b_1 + b_2) \times h \)
Applying the Trapezium Area Formula
In this question, we are given the following information about the trapezium:
| Parameter | Value |
|---|---|
| Area of the trapezium | 18 sq. cm |
| Height (h) | 3 cm |
| One base (parallel side, let's say \(b_1\)) | 5 cm |
We need to find the length of the other parallel side (\(b_2\)). We can use the area formula and substitute the given values:
\( 18 = \frac{1}{2} \times (5 + b_2) \times 3 \)
Step-by-Step Calculation to Find the Parallel Side
Now, let's solve the equation for \(b_2\):
- Multiply both sides of the equation by 2 to remove the fraction:
\( 18 \times 2 = (5 + b_2) \times 3 \)
\( 36 = (5 + b_2) \times 3 \)
- Divide both sides of the equation by 3:
\( \frac{36}{3} = 5 + b_2 \)
\( 12 = 5 + b_2 \)
- Subtract 5 from both sides to isolate \(b_2\):
\( 12 - 5 = b_2 \)
\( 7 = b_2 \)
So, the length of the side parallel to the base is 7 cm.
Conclusion
Using the formula for the area of a trapezium and the given values for area, height, and one parallel side, we calculated the length of the other parallel side to be 7 cm.
Revision Table: Trapezium Properties
| Property | Description | Formula (if applicable) |
|---|---|---|
| Shape | Quadrilateral with at least one pair of parallel sides | - |
| Parallel Sides (Bases) | The pair of opposite sides that are parallel | \(b_1, b_2\) |
| Height | The perpendicular distance between the parallel sides | \(h\) |
| Area | The space enclosed within the trapezium | \( \text{Area} = \frac{1}{2}(b_1 + b_2)h \) |
| Perimeter | Sum of the lengths of all four sides | \( \text{Perimeter} = \text{side}_1 + \text{side}_2 + \text{side}_3 + \text{side}_4 \) |
Additional Information on Trapeziums
- Trapeziums can be classified based on their non-parallel sides and angles. An isosceles trapezium has non-parallel sides of equal length and base angles that are equal.
- A right trapezium has at least one right angle (one of the non-parallel sides is perpendicular to the parallel sides).
- A parallelogram is a special case of a trapezium where both pairs of opposite sides are parallel. In this case, the height is the perpendicular distance between one pair of parallel sides, and the area formula simplifies to base × height.
Important Questions from Quadrilaterals
The area of a rectangle is 300 cm 2and the length of its diagonal is 25 cm. The perimeter of the rectangle (in cm) is:
The adjacent angles of a rhombus are in the ratio of 3 : 6. The smallest angle of the rhombus is:
Find the area of a rhombus whose perimeter is 116 cm and one diagonal is 42 cm in length.
The ratio of sides to a quadrilateral is 2 ∶ 3 ∶ 4 ∶ 5 and the perimeter is 560 cm. Find out the smallest side. (In cm)
The ratio of arms of a quadrilateral is 2 ∶ 3 ∶ 4 ∶ 5 and the Perimeter is 616 cm. Find out the smallest arm. (In cm)
