Introduction to the Inradius of a Triangle

The concept of the incircle of a triangle is fundamental in geometry. The incircle, or inscribed circle, is a circle that is tangent to all three sides of the triangle. The point of tangency, known as the incenter, is the center of the incircle and is the point of concurrency of the angle bisectors of the triangle. The radius of the incircle, or inradius, is a crucial metric in many geometric calculations and applications.

How to Find the Inradius of a Triangle Given Side Lengths

Giving you the side lengths of a triangle allows for an in-depth calculation of the inradius. Let's consider a triangle with side lengths of 8 cm, 6 cm, and 10 cm. By following a systematic approach, we can accurately determine the inradius.

Step-by-Step Calculation

One method to find the inradius involves the following steps:

Calculate the Semi-Perimeter: The first step is to calculate the semi-perimeter of the triangle. The semi-perimeter, denoted as s, is half the perimeter of the triangle. Calculate the Area: The area of the triangle can be calculated using Heron's formula, which involves the semi-perimeter and the side lengths. Calculate the Inradius: Finally, the inradius can be calculated by dividing the area of the triangle by the semi-perimeter.

Example Calculation

Let's proceed with the calculation using the triangle with side lengths of 8 cm, 6 cm, and 10 cm.

Step 1: Calculate the Semi-Perimeter
s (a b c) / 2
s (8 6 10) / 2
s 24 / 2
s 12 cm

Step 2: Calculate the Area
A √[s(s-a)(s-b)(s-c)]
A √[12(12-8)(12-6)(12-10)]
A √[12(4)(6)(2)]
A √[12 * 4 * 6 * 2]
A √[576]
A 24 cm2

Step 3: Calculate the Inradius
r A / s
r 24 / 12
r 2 cm

Therefore, the inradius of the triangle with side lengths of 8 cm, 6 cm, and 10 cm is 2 cm.

Alternative Methods and Insights

There are alternative methods to determine the inradius, such as the use of a right triangle's properties. Given that a 8 cm, 6 cm, and 10 cm triangle is a right triangle (where 10 is the hypotenuse), the inradius can be calculated using a simpler formula:

Right Triangle Shortcut

For a right triangle, the inradius can be calculated using the formula:

r (a * b - c) / 2, where a and b are the legs and c is the hypotenuse.

Therefore, for the given right triangle:

r (8 * 6 - 10) / 2
r (48 - 10) / 2
r 38 / 2
r 19 / 2
r 9.5 / 2
r 2 cm

This confirms the earlier calculation and provides a more geometric insight into the problem.

Visual Representation and Diagram

Historically, visual diagrams have been helpful in understanding geometric concepts. In the case of the incircle, the incenter is equidistant from all three sides of the triangle, forming a right angle at the point of tangency. A right-angled triangle with a square inscribed within it can demonstrate the relationship between the inradius, the sides, and the tangents.

Visual Diagram for a Right Triangle

A 8 cm, 6 cm, and 10 cm right triangle with an inscribed square of side length r.

The square with side length r touches each side of the triangle, forming right angles at the points of tangency. This visual representation helps in understanding the congruence of the tangents and the relationship between the inradius and the sides of the triangle.

Conclusion

The inradius of a triangle with side lengths of 8 cm, 6 cm, and 10 cm is 2 cm, as demonstrated through both the Heron's formula method and the right triangle shortcut. Understanding the properties and methods for calculating the inradius is essential for solving a variety of geometric problems and applications in mathematics.