ABCD is a quadrilateral. The diagonals of ABCD intersect at the point P. The area of the triangles APD and BPC are 27 and 12, respectively. If the areas of the triangles APB and CPD are equal, then the area of triangle APB is

Options

A. 12
B. 18
C. 15
D. 16

Answer

To find the area of triangle APB in the quadrilateral ABCD, we can use the information given about the areas of the triangles formed by the diagonals intersecting at point P.

### Step-by-Step Solution:

1. **Identify the Areas Given:**
- Area of triangle APD = 27
- Area of triangle BPC = 12
- Let the area of triangle APB = x
- Since the areas of triangles APB and CPD are equal, we can denote the area of triangle CPD also as x.

2. **Set Up the Area Relationship:**
- The total area of quadrilateral ABCD can be expressed as:
\[ \text{Area of ABCD} = \text{Area of APD} + \text{Area of BPC} + \text{Area of APB} + \text{Area of CPD \]
- Substituting the known values:
\[ \text{Area of ABCD} = 27 + 12 + x + x = 39 + 2x \]

3. **Using the Property of Similar Triangles:**
- Since triangles APB and CPD are equal in area and the triangles APD and BPC are given, we can use the ratio of the areas:
\[ \frac{\text{Area of APD}}{\text{Area of BPC}} = \frac{27}{12} = \frac{9}{4} \]
- This ratio can also be expressed in terms of the areas of triangles APB and CPD:
\[ \frac{x}{x} = 1 \]
- Therefore, we can set up the equation based on the areas:
\[ \frac{27}{12} = \frac{x}{x} \]
- This means that the areas of triangles APB and CPD are proportional to the areas of triangles APD and BPC.

4. **Finding the Value of x:**
- From the ratio of the areas, we can express:
\[ \frac{27}{12} = \frac{x}{x} \]
- This does not provide new information since x cancels out. Instead, we can use the relationship derived from the total area:
- We know that the total area can also be expressed in terms of the areas of triangles APD and BPC:
\[ \text{Area of ABCD} = \text{Area of APD} + \text{Area of BPC} + 2x \]
- Setting the two expressions for the area of ABCD equal:
\[ 39 + 2x = 27 + 12 + 2x \]
- Simplifying gives:
\[ 39 + 2x = 39 + 2x \]
- This confirms our setup is correct.

5. **Final Calculation:**
- Since we know the areas of triangles APD and BPC, we can find x using the area ratios:
\[ 27 + 12 = 39 \]
- The area of triangle APB can be calculated as:
\[ x = \sqrt{27 \times 12} = \sqrt{324} = 18 \]

### Conclusion:
The area of triangle APB is **18**.