Solving Geometric Problems: Trapezium Area and Altitude Calculation

In this article, we will explore the geometric problem involving the area and altitude of a trapezium. We'll break down the steps to find the lengths of the parallel sides using the given information. This problem serves as a practical example of applying geometric formulas and algebraic calculations.

Problem Statement

The area and altitude of a trapezium are given as 30 cm2 and 4 cm, respectively. The problem also states that one parallel side is double the length of the other. Our task is to calculate the lengths of the parallel sides.

Given Information and Initial Setup

Let's denote the shorter parallel side as a and the longer side as 2a. The formula for the area of a trapezium can be expressed as:

( text{Area} frac{1}{2} times (text{sum of parallel sides}) times text{altitude} )

Substituting the given values, we have:

( 30 frac{1}{2} times (a 2a) times 4 )

Let's simplify this equation step by step:

Step-by-step Solution

1. **Sum of Parallel Sides**: Combine the parallel sides. Since one is double the other, the sum is (a 2a 3a).

2. **Substitute and Simplify**: Substitute these values into the area formula:

( 30 frac{1}{2} times 3a times 4 )

3. **Simplify the Equation**: Multiply the terms inside the parentheses:

( 30 2 times 3a )

4. **Further Simplification**: Simplify the right-hand side:

( 30 6a )

5. **Solve for a**: Divide both sides by 6 to find the value of a:

( a frac{30}{6} 5 )

Final Answer and Verification

The shorter parallel side is 5 cm, and the longer side is (2 times 5 10) cm. This solution meets the requirements of the problem statement.

Conclusion

This problem demonstrates the application of basic geometric formulas and algebraic manipulations. Understanding these concepts and techniques is crucial for solving more complex geometric problems.

Key Concepts

Throughout the problem, we utilized the following key concepts:

Trapezium Area Formula: (text{Area} frac{1}{2} times (text{sum of parallel sides}) times text{altitude}) Algebraic Manipulation: Solving equations to find unknown values. Geometric Reasoning: Applying logical steps to break down and solve the problem.

Application in Real-world Scenarios

Geometric solving techniques, such as the one demonstrated in this problem, are applicable in various fields, including engineering, architecture, and design. Understanding how to calculate the area and dimensions of shapes is crucial for professionals in these fields.

Additional Resources

To further enhance your understanding of these concepts, consider exploring the following resources:

Textbooks and Online Courses: Look for comprehensive guides on geometry and algebra. Practice Problems: Solve additional geometry problems available online or in textbooks. Tutoring and Mentorship: Seek guidance from experienced educators or mentors.