The area of a triangle ABC is `10.8 cm^(2)`. If CP = PB and 2AQ = QB, then the area of the triangle APO Is  

Options

A. `3.6 cm^(2)`
B. `0.9 cm^(2)`
C. `2.7 cm^(2)`
D. `1.8 cm^(2)`

Answer

To find the area of triangle AOP given that the area of triangle ABC is 10.8 cm², we will follow these steps:

### Step-by-Step Solution:

1. **Understanding the Given Information**:
- We know that the area of triangle ABC is 10.8 cm².
- The points P and Q divide the segments BC and AB respectively. Specifically, CP = PB and 2AQ = QB.

2. **Setting Up the Ratios**:
- Since CP = PB, we can denote the lengths as follows:
- Let CP = x, then PB = x (thus, BC = 2x).
- For AQ and QB, since 2AQ = QB, we can denote:
- Let AQ = y, then QB = 2y (thus, AB = 3y).

3. **Finding the Area of Triangle AQP**:
- The area of triangle AQP can be calculated based on the ratios of the segments.
- The area of triangle AQP is proportional to the segments AQ and QB.
- Since AQ:QB = 1:2, the area of triangle AQP will be 1/3 of the area of triangle ABC.

4. **Calculating Area of Triangle AQP**:
- Area of triangle AQP = (1/3) * Area of triangle ABC
- Area of triangle AQP = (1/3) * 10.8 cm² = 3.6 cm².

5. **Finding the Area of Triangle AOP**:
- The area of triangle AOP is half of the area of triangle AQP because point O is the midpoint of segment AP.
- Therefore, Area of triangle AOP = (1/2) * Area of triangle AQP
- Area of triangle AOP = (1/2) * 3.6 cm² = 1.8 cm².

### Final Answer:
The area of triangle AOP is **1.8 cm²**.

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