The text in the image is in Kazakh and translates to:

7. The radius of the inscribed circle of a triangle is 3, and the perimeter of the triangle is 20. Find the area of the triangle.

Solution:

The area ($A$) of a triangle with an inscribed circle can be calculated using the formula:

$A = r \cdot s$

Where:

• $r$ is the radius of the inscribed circle.
• $s$ is the semiperimeter of the triangle.

Step 1: Calculate the semiperimeter ($s$)

$s = \frac{\text{perimeter}}{2} = \frac{20}{2} = 10$

Step 2: Substitute into the formula

$A = r \cdot s = 3 \cdot 10 = 30$

Final Answer:

The area of the triangle is 30 square units.

Would you like a detailed explanation of any part?

Related Questions:

1. What is the relationship between the radius of the inscribed circle and the area of a triangle?
2. How do you calculate the semiperimeter of a triangle if given the side lengths?
3. Can the formula $A = r \cdot s$ be derived from other geometric properties?
4. What are the formulas for triangles without an inscribed circle?
5. How do the dimensions of a triangle affect its inscribed circle?

Tip:

For any triangle, the inscribed circle radius is always connected to the triangle's area and semiperimeter. Knowing this simplifies many problems.