The difference between two parallel sides of a trapezium is 4 cm. The perpendicular distance between them is 19 cm. If the area of the trapezium is 475 `cm^(2)`. Find the lengths (cm) of the parallel sides:

Options

A. 27, 23
B. 27, 24
C. 27, 31
D. 29, 25

Answer

To solve the problem, we need to find the lengths of the parallel sides of the trapezium given the following information:

1. The difference between the two parallel sides is 4 cm.
2. The perpendicular distance between the parallel sides is 19 cm.
3. The area of the trapezium is 475 cm².

Let's denote the lengths of the two parallel sides as:
- \( a \) (the shorter side)
- \( b \) (the longer side)

From the problem, we know:
- \( b - a = 4 \) (Equation 1)
- The area of the trapezium can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \times (a + b) \times h \]
where \( h \) is the height (perpendicular distance between the parallel sides). Here, \( h = 19 \) cm.

Substituting the values into the area formula, we have:
\[ 475 = \frac{1}{2} \times (a + b) \times 19 \]

Now, let's simplify this equation step by step.

### Step 1: Multiply both sides by 2 to eliminate the fraction.
\[ 950 = (a + b) \times 19 \]

**Hint:** To simplify equations, multiplying or dividing both sides by the same number can help eliminate fractions.

### Step 2: Divide both sides by 19 to isolate \( a + b \).
\[ a + b = \frac{950}{19} \]
Calculating the right side:
\[ a + b = 50 \]

**Hint:** When isolating a variable, performing inverse operations (like division) can help you find the value you need.

### Step 3: Now we have two equations:
1. \( b - a = 4 \) (Equation 1)
2. \( a + b = 50 \) (Equation 2)

### Step 4: Solve these equations simultaneously. From Equation 1, we can express \( b \) in terms of \( a \):
\[ b = a + 4 \]

**Hint:** Substituting one equation into another can help simplify the problem.

### Step 5: Substitute \( b \) in Equation 2:
\[ a + (a + 4) = 50 \]
This simplifies to:
\[ 2a + 4 = 50 \]

### Step 6: Subtract 4 from both sides.
\[ 2a = 46 \]

### Step 7: Divide both sides by 2 to find \( a \).
\[ a = 23 \]

### Step 8: Now substitute \( a \) back into the expression for \( b \):
\[ b = 23 + 4 = 27 \]

### Conclusion:
The lengths of the parallel sides are:
- \( a = 23 \) cm
- \( b = 27 \) cm

**Final Answer:** The lengths of the parallel sides of the trapezium are 23 cm and 27 cm.