The area of a quadrilateal is 342 sq. m. The perpendiculars from two of its opposite vertices to the diagonal are 12m and 12m. What is the length of the diagonal?

Options

A. 28.6m
B. 25.3m
C. 28.5m
D. 22.5m

Answer

To find the length of the diagonal of the quadrilateral given the area and the heights from two opposite vertices, we can follow these steps:

### Step 1: Understand the Area of the Quadrilateral
The area of a quadrilateral can be divided into two triangles by drawing a diagonal. The area of each triangle can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]

### Step 2: Set Up the Problem
Let the quadrilateral be \(ABCD\) with diagonal \(BD\). The perpendiculars from vertices \(A\) and \(C\) to the diagonal \(BD\) are both 12 m. Therefore, we denote:
- Height from \(A\) to \(BD\) as \(AM = 12\) m
- Height from \(C\) to \(BD\) as \(CN = 12\) m

### Step 3: Write the Area Equation
The area of the quadrilateral can be expressed as the sum of the areas of triangles \(ABD\) and \(CDB\):
\[ \text{Area}_{ABD} + \text{Area}_{CDB} = 342 \text{ m}^2 \]

Using the area formula for both triangles:
\[ \frac{1}{2} \times DB \times AM + \frac{1}{2} \times DB \times CN = 342 \]

### Step 4: Substitute the Heights
Substituting the heights \(AM\) and \(CN\) into the equation:
\[ \frac{1}{2} \times DB \times 12 + \frac{1}{2} \times DB \times 12 = 342 \]

### Step 5: Simplify the Equation
This simplifies to:
\[ \frac{1}{2} \times DB \times (12 + 12) = 342 \]
\[ \frac{1}{2} \times DB \times 24 = 342 \]

### Step 6: Solve for \(DB\)
To isolate \(DB\), multiply both sides by 2:
\[ DB \times 24 = 684 \]
Now, divide both sides by 24:
\[ DB = \frac{684}{24} \]

### Step 7: Perform the Division
Calculating the division:
\[ DB = 28.5 \text{ m} \]

### Conclusion
Thus, the length of the diagonal \(BD\) is \(28.5\) meters.

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