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The problem in the image asks to determine the area of an isosceles triangle \(ABC\) with an inscribed circle of radius 7 cm. It is given that angle \(A\) is 60°.

Let's solve this step by step.

### Step 1: Understand the given information
- The triangle \(ABC\) is isosceles.
- The circle inscribed within the triangle has a radius \(r = 7 \, \text{cm}\).
- Angle \( \angle A = 60^\circ \).

### Step 2: Use the formula for the area of a triangle with an inscribed circle
The area \( \text{Area} \) of a triangle can be calculated using the formula:

\[
\text{Area} = r \times s
\]

where:
- \( r \) is the radius of the inscribed circle.
- \( s \) is the semi-perimeter of the triangle.

### Step 3: Relate the given angle with the triangle's properties
Given that \( \angle A = 60^\circ \) and the triangle is isosceles, the triangle is not only isosceles but also equilateral. For an equilateral triangle:

\[
\text{Semi-perimeter} \, s = \frac{3a}{2}
\]
\[
\text{Area} = \frac{\sqrt{3}}{4} a^2
\]
\[
r = \frac{a\sqrt{3}}{6}
\]

Given \( r = 7 \, \text{cm} \):

\[
7 = \frac{a\sqrt{3}}{6} \implies a = \frac{42}{\sqrt{3}} \implies a = 14\sqrt{3} \, \text{cm}
\]

### Step 4: Calculate the area
Substituting the value of \( a \):

\[
\text{Area} = \frac{\sqrt{3}}{4} \times (14\sqrt{3})^2
\]
\[
\text{Area} = \frac{\sqrt{3}}{4} \times 588
\]
\[
\text{Area} = \frac{588\sqrt{3}}{4} = 147\sqrt{3} \approx 254.56 \, \text{cm}^2
\]

### Final Answer:
The area of the triangle is approximately **254.6 cm²** when rounded to 1 decimal place.

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Would you like to go over any particular step in more detail, or do you have any questions?

Here are some related questions you might be interested in:
1. How do you derive the formula for the radius of the inscribed circle in an equilateral triangle?
2. Can this problem be solved using trigonometry without assuming the triangle is equilateral?
3. What would change if angle \( \angle A \) were different from 60°?
4. How would the problem be solved if the triangle was not isosceles?
5. Can we find the side lengths of the triangle directly from the given radius without using the area formula?

**Tip:** In geometric problems, always consider whether special triangle properties (like equilateral or right-angle) simplify the problem.

Determine the area of an isosceles triangle ABC with an inscribed circle of radius 7 cm, where angle A is 60°. Give your answer correct to 1 decimal place.

This detailed solution guides you through calculating the area of an isosceles triangle with an inscribed circle of radius 7 cm, given that one angle is 60°. Ideal for Grades 10-12 students, this problem covers key geometry concepts, including properties of equilateral triangles and inscribed circles.

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