The problem states that the area of a triangle is $84 \, \text{cm}^2$ and its perimeter is $72 \, \text{cm}$. We need to find the radius of the inscribed circle (inradius) of the triangle.

Formula for the inradius ($r$) of a triangle:

The formula to calculate the inradius is: $r = \frac{A}{s}$ where:

• $A$ is the area of the triangle.
• $s$ is the semi-perimeter of the triangle, calculated as $s = \frac{\text{Perimeter}}{2}$.

Step 1: Calculate the semi-perimeter

Given: $\text{Perimeter} = 72 \, \text{cm}$ $s = \frac{72}{2} = 36 \, \text{cm}$

Step 2: Use the formula to find the inradius

Given the area: $A = 84 \, \text{cm}^2$ $r = \frac{84}{36} = 2.33 \, \text{cm}$

Conclusion

The radius of the inscribed circle of the triangle is $2.33 \, \text{cm}$.

Would you like more details or have any questions about this solution?

Here are 5 related questions for further exploration:

1. How do you derive the formula for the inradius of a triangle?
2. What is the relationship between the inradius and the area of a triangle?
3. Can the formula for inradius be applied to all types of triangles?
4. How do you find the inradius if you know the lengths of all three sides of the triangle?
5. What is the significance of the inradius in geometric constructions?

Tip: Remember that the inradius formula works for any triangle, but knowing the semi-perimeter is crucial!