A trapezium with an area of `77 cm^2` has a height of 7 cm. If the difference between the lengths of its parallel sides is 4 cm, find the length of the longer side.

Options

A. 13 cm
B. 16 cm
C. 19 cm
D. 22 cm

Answer

To solve the problem step by step, we will use the formula for the area of a trapezium and the information provided in the question.

### Step 1: Write down the formula for the area of 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, and \( h \) is the height.

### Step 2: Substitute the known values into the area formula.
We know:
- Area \( A = 77 \, \text{cm}^2 \)
- Height \( h = 7 \, \text{cm} \)

Substituting these values into the formula gives:
\[ 77 = \frac{1}{2} \times (b_1 + b_2) \times 7 \]

### Step 3: Simplify the equation.
Multiply both sides by 2 to eliminate the fraction:
\[ 154 = (b_1 + b_2) \times 7 \]
Now, divide both sides by 7:
\[ b_1 + b_2 = \frac{154}{7} = 22 \]

### Step 4: Set up the equation for the difference between the parallel sides.
We are given that the difference between the lengths of the parallel sides is 4 cm. We can express this as:
\[ b_2 - b_1 = 4 \]
where \( b_2 \) is the longer side and \( b_1 \) is the shorter side.

### Step 5: Solve the system of equations.
Now we have two equations:
1. \( b_1 + b_2 = 22 \)
2. \( b_2 - b_1 = 4 \)

We can solve these equations simultaneously.

From the second equation, we can express \( b_2 \) in terms of \( b_1 \):
\[ b_2 = b_1 + 4 \]

Now substitute \( b_2 \) in the first equation:
\[ b_1 + (b_1 + 4) = 22 \]
This simplifies to:
\[ 2b_1 + 4 = 22 \]
Subtract 4 from both sides:
\[ 2b_1 = 18 \]
Now divide by 2:
\[ b_1 = 9 \]

### Step 6: Find \( b_2 \).
Now substitute \( b_1 \) back into the equation for \( b_2 \):
\[ b_2 = b_1 + 4 = 9 + 4 = 13 \]

### Conclusion: State the length of the longer side.
Thus, the length of the longer side \( b_2 \) is:
\[ \boxed{13 \, \text{cm}} \]