To solve the problem of finding the distance between the parallel sides of a trapezium (also known as a trapezoid in some regions), we can use the formula for the area of a trapezium. Let's break down the steps clearly. ### Step 1: Understand the Area Formula for a Trapezium The area \( A \) of a trapezium can be calculated using the formula: \[ A = \frac{1}{2} \times (b_1 + b_2) \times h \] where: - \( b_1 \) and \( b_2 \) are the lengths of the two parallel sides, - \( h \) is the height (or distance) between the parallel sides. ### Step 2: Identify the Given Values From the problem, we have: - \( b_1 = 5 \) cm (length of the shorter parallel side), - \( b_2 = 9 \) cm (length of the longer parallel side), - \( A = 21 \) cm² (area of the trapezium). ### Step 3: Substitute the Values into the Area Formula We can rearrange the area formula to solve for \( h \): \[ h = \frac{2A}{b_1 + b_2} \] Now, substituting the known values into the formula: \[ h = \frac{2 \times 21}{5 + 9} \] ### Step 4: Calculate the Denominator First, calculate \( b_1 + b_2 \): \[ b_1 + b_2 = 5 + 9 = 14 \] ### Step 5: Substitute Back into the Height Formula Now substitute this back into the equation for \( h \): \[ h = \frac{2 \times 21}{14} \] ### Step 6: Perform the Multiplication and Division Calculate the numerator: \[ 2 \times 21 = 42 \] Now divide by 14: \[ h = \frac{42}{14} = 3 \] ### Conclusion: The Distance Between the Parallel Sides Thus, the distance between the parallel sides of the trapezium is: \[ h = 3 \text{ cm} \] ### Step 7: Evaluate the Options Now, let's look at the options provided: - A. 3 cm (Correct) - B. 4 cm (Incorrect) - C. 6 cm (Incorrect) - D. 7 cm (Incorrect) ### Explanation of Incorrect Options - **Option B (4 cm)**: This value does not satisfy the area calculation based on the given lengths of the parallel sides. If we substitute \( h = 4 \) into the area formula, we would get a larger area than 21 cm². - **Option C (6 cm)**: Similar to option B, substituting \( h = 6 \) would yield an area greater than 21 cm², which is not consistent with the problem statement. - **Option D (7 cm)**: Again, substituting \( h = 7 \) would result in an area that exceeds 21 cm², making this option invalid. ### Revision Summary - The area of a trapezium is calculated using the formula \( A = \frac{1}{2} \times (b_1 + b_2) \times h \). - To find the height \( h \), rearrange the formula to \( h = \frac{2A}{b_1 + b_2} \). - Substitute the known values into the formula to find \( h \). - The correct answer is 3 cm, as it satisfies the area condition given in the problem. This thorough breakdown should help you understand how to approach similar problems involving trapeziums and their areas!