ABC is an equilateral triangle . P and Q are two points on AB and AC such that `PQ|\|BC. PQ=5 cm`. Find the area of `Delta APQ`

Answer

To find the area of triangle APQ given that ABC is an equilateral triangle and PQ is parallel to BC with a length of 5 cm, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Triangle**:
- We have triangle ABC which is equilateral, meaning all sides are equal and all angles are 60 degrees.
- Points P and Q are located on sides AB and AC respectively, such that line segment PQ is parallel to side BC.

2. **Identifying Angles**:
- Since PQ is parallel to BC, the angles formed at points A, P, and Q are equal to the corresponding angles at points A, B, and C.
- Therefore, angle APQ = angle ABC = 60 degrees and angle AQP = angle ACB = 60 degrees.

3. **Determining Triangle Type**:
- Since all angles in triangle APQ are 60 degrees, triangle APQ is also an equilateral triangle.

4. **Using the Length of PQ**:
- The length of PQ is given as 5 cm. In an equilateral triangle, all sides are equal, so the lengths of AP and AQ are also 5 cm.

5. **Calculating the Area**:
- The formula for the area of an equilateral triangle is given by:
\[ \text{Area} = \frac{\sqrt{3}}{4} \times \text{side}^2 \]
- Substituting the length of the side (which is 5 cm) into the formula:
\[ \text{Area} = \frac{\sqrt{3}}{4} \times (5)^2 = \frac{\sqrt{3}}{4} \times 25 = \frac{25\sqrt{3}}{4} \text{ cm}^2 \]

### Final Answer:
The area of triangle APQ is \(\frac{25\sqrt{3}}{4} \text{ cm}^2\).