The area of a quadrilateral is `360m^(2)` and the perpendiculars drawn to one of the diagonal from the opposite vertices are 10m and 8m. Find the length of the diagonal.

Answer

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

### Step 1: Understand the Problem
We are given:
- Area of the quadrilateral = 360 m²
- Perpendiculars from opposite vertices to the diagonal:
- Length of perpendicular from vertex B (BK) = 10 m
- Length of perpendicular from vertex D (DM) = 8 m

We need to find the length of the diagonal AC.

### Step 2: Set Up the Area Equation
The area of a quadrilateral can be expressed as the sum of the areas of two triangles formed by the diagonal. Thus, we can write:

\[ \text{Area of Quadrilateral} = \text{Area of } \triangle ABC + \text{Area of } \triangle ADC \]

### Step 3: Calculate the Areas of the Triangles
The area of a triangle can be calculated using the formula:

\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]

For both triangles, the base is the length of the diagonal AC. Therefore, we can express the areas as:

\[ \text{Area of } \triangle ABC = \frac{1}{2} \times AC \times BK = \frac{1}{2} \times AC \times 10 \]

\[ \text{Area of } \triangle ADC = \frac{1}{2} \times AC \times DM = \frac{1}{2} \times AC \times 8 \]

### Step 4: Combine the Areas
Now we can combine these areas:

\[ 360 = \frac{1}{2} \times AC \times 10 + \frac{1}{2} \times AC \times 8 \]

### Step 5: Factor Out the Common Terms
Factoring out \(\frac{1}{2} \times AC\):

\[ 360 = \frac{1}{2} \times AC \times (10 + 8) \]

This simplifies to:

\[ 360 = \frac{1}{2} \times AC \times 18 \]

### Step 6: Solve for AC
To isolate AC, multiply both sides by 2:

\[ 720 = AC \times 18 \]

Now, divide both sides by 18:

\[ AC = \frac{720}{18} \]

Calculating this gives:

\[ AC = 40 \, \text{m} \]

### Conclusion
The length of the diagonal AC is **40 meters**.

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