The area of a trapezium is 475 `cm^(2)` and its height is 19 cm. Find the lengths of its two parallel sides if one side is 4 cm greater than the other.

Explanation

To solve the problem, we will use the formula for the area of a trapezium:

\[ \text{Area} = \frac{1}{2} \times (a + b) \times h \]

where \( a \) and \( b \) are the lengths of the two parallel sides, and \( h \) is the height.

### Step 1: Identify the variables
Let:
- \( x \) = length of the shorter parallel side
- \( x + 4 \) = length of the longer parallel side (since it is 4 cm greater than the shorter side)

### Step 2: Write the area formula
Given that the area of the trapezium is 475 cm² and the height is 19 cm, we can substitute these values into the area formula:

\[ 475 = \frac{1}{2} \times (x + (x + 4)) \times 19 \]

### Step 3: Simplify the equation
First, simplify the expression inside the parentheses:

\[ x + (x + 4) = 2x + 4 \]

Now, substitute this back into the area equation:

\[ 475 = \frac{1}{2} \times (2x + 4) \times 19 \]

### Step 4: Eliminate the fraction
Multiply both sides by 2 to eliminate the fraction:

\[ 950 = (2x + 4) \times 19 \]

### Step 5: Divide by 19
Now, divide both sides by 19:

\[ \frac{950}{19} = 2x + 4 \]

Calculating \( \frac{950}{19} \):

\[ 50 = 2x + 4 \]

### Step 6: Solve for \( x \)
Now, subtract 4 from both sides:

\[ 50 - 4 = 2x \]

\[ 46 = 2x \]

Now, divide by 2:

\[ x = 23 \]

### Step 7: Find the lengths of the parallel sides
Now that we have \( x \), we can find the lengths of the two parallel sides:

- Shorter side \( a = x = 23 \) cm
- Longer side \( b = x + 4 = 23 + 4 = 27 \) cm

### Final Answer
The lengths of the two parallel sides are:
- Shorter side: 23 cm
- Longer side: 27 cm

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