Question

Find the radius of the circle that is inscribed in a triangle with sides of lengths 3,4 , and 5 (see Figure 21).

Short answer

The radius of the inscribed circle is 1 unit.

Step-by-step solution

Identify the Triangle Type

Recognize the triangle with sides 3, 4, and 5. Since it satisfies the Pythagorean theorem, \(3^2 + 4^2 = 5^2\), it is a right triangle.

Use the Right Triangle Area Formula

Calculate the area \(A\) of the triangle using the formula for the area of a right triangle: \(A = \frac{1}{2} \times \text{base} \times \text{height}\). Here, let the base be 3 and the height be 4: \[A = \frac{1}{2} \times 3 \times 4 = 6.\]

Calculate the Semi-Perimeter

Find the semi-perimeter \(s\), which is given by \(s = \frac{a + b + c}{2}\). For the triangle with sides 3, 4, and 5: \[s = \frac{3 + 4 + 5}{2} = 6.\]

Apply the Inradius Formula

Use the inradius formula for triangles, which is \( r = \frac{A}{s} \). Substitute the known values into the formula \[r = \frac{6}{6} = 1.\]

Conclusion: Interpretation of Results

The radius of the inscribed circle (incircle) in this triangle is 1 unit.

Key concepts

Right Triangle

Understanding what a right triangle is can help simplify various calculations. A right triangle, one of the fundamental types of triangles, has a special quality: one of its angles is exactly 90 degrees.
This specific characteristic makes right triangles unique in geometry.The sides of the right triangle include:

• The hypotenuse, which is the longest side and is opposite the right angle.
• Two legs, which are the other two sides forming the right angle.

A right triangle with side lengths 3, 4, and 5 is known as a Pythagorean triple.
This is because these numbers satisfy the Pythagorean theorem: \[3^2 + 4^2 = 5^2\] This makes the calculation of other properties, like area and inradius, straightforward.

Semi-Perimeter

The semi-perimeter of a triangle is an intermediate step used in various geometric formulas, such as the inradius formula.
It is half the perimeter of the triangle.To find the semi-perimeter, simply add the lengths of all the sides of the triangle and divide by two:\[s = \frac{a + b + c}{2}\]For example, in a triangle with side lengths 3, 4, and 5, the semi-perimeter is calculated as follows:\[s = \frac{3 + 4 + 5}{2} = 6\]Understanding the semi-perimeter is important because it often serves as a bridge in calculations involving other geometric properties, like the area or the inradius.

Inradius Formula

The inradius of a triangle is the radius of the largest circle that fits inside the triangle. This circle touches each side of the triangle exactly once.To calculate the inradius \( r \), use the formula:\[r = \frac{A}{s}\]where \( A \) is the area of the triangle and \( s \) is the semi-perimeter.
This simple formula works for any triangle.For a right triangle with side lengths 3, 4, and 5, we have already calculated:

• The area \( A = 6 \)
• The semi-perimeter \( s = 6 \)

Using the inradius formula, the inradius is:\[r = \frac{6}{6} = 1\]Thus, the radius of the inscribed circle is 1 unit.

Area of a Triangle

Calculating the area of a triangle is a basic geometric task. For right triangles, this calculation is even easier because you can use the standard formula:\[A = \frac{1}{2} \times \text{base} \times \text{height}\]In this context, the base and height are the two legs that form the right angle.For a right triangle with a base of 3 and a height of 4, the area is:\[A = \frac{1}{2} \times 3 \times 4 = 6\]Knowing how to find the area helps in various applications, such as computing the semi-perimeter, the inradius, and other critical geometric properties.With practice, determining the area of right triangles becomes second nature. Make sure to work through several examples to solidify this concept.