The two parallel sides and the distance between them are in the ratio `3: 4: 2`. If the area of the trapezium is 175 `cm^(2)`, find its height.

Answer

To solve the problem step by step, we will follow these instructions:

### Step 1: Understand the Ratios
The problem states that the two parallel sides and the distance between them (height) are in the ratio 3:4:2. We can denote:
- The first parallel side as \(3x\)
- The second parallel side as \(4x\)
- The height (distance between the parallel sides) as \(2x\)

### Step 2: Write the Area Formula
The area \(A\) of a trapezium is given by the formula:
\[ A = \frac{1}{2} \times (b_1 + b_2) \times h \]
where \(b_1\) and \(b_2\) are the lengths of the parallel sides, and \(h\) is the height.

### Step 3: Substitute the Values
From the problem, we know the area \(A = 175 \, cm^2\). Substituting the values we defined:
\[ 175 = \frac{1}{2} \times (3x + 4x) \times (2x) \]

### Step 4: Simplify the Equation
Now simplify the equation:
\[ 175 = \frac{1}{2} \times (7x) \times (2x) \]
The \(2\) in the denominator cancels with the \(2\) in the height:
\[ 175 = 7x^2 \]

### Step 5: Solve for \(x^2\)
Now, isolate \(x^2\):
\[ x^2 = \frac{175}{7} \]
Calculating the right side:
\[ x^2 = 25 \]

### Step 6: Find \(x\)
Taking the square root of both sides:
\[ x = \sqrt{25} = 5 \, cm \]

### Step 7: Calculate the Height
Now, we can find the height \(h\):
\[ h = 2x = 2 \times 5 = 10 \, cm \]

### Final Answer
The height of the trapezium is \(10 \, cm\).

---