What is the inradius of a triangle with side lengths 7, 24, 25?

To find the inradius (radius of the inscribed circle) of a triangle with given side lengths, we can use the following formula:

\[
r = \frac{A}{s}
\]

Where:
- \( r \) is the inradius
- \( A \) is the area of the triangle
- \( s \) is the semi-perimeter of the triangle

### Step 1: Calculate the semi-perimeter \( s \)
The semi-perimeter is given by:

\[
s = \frac{a + b + c}{2}
\]

For the triangle with sides \( a = 7 \), \( b = 24 \), and \( c = 25 \):

\[
s = \frac{7 + 24 + 25}{2} = \frac{56}{2} = 28
\]

### Step 2: Calculate the area \( A \)
Since the triangle has sides 7, 24, and 25, we recognize it as a right triangle (because \( 7^2 + 24^2 = 25^2 \)).

The area \( A \) of a right triangle is given by:

\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]

Here, we can take \( 7 \) as the base and \( 24 \) as the height:

\[
A = \frac{1}{2} \times 7 \times 24 = \frac{1}{2} \times 168 = 84
\]

### Step 3: Calculate the inradius \( r \)
Now, use the formula \( r = \frac{A}{s} \):

\[
r = \frac{84}{28} = 3
\]

So, the inradius \( r \) of the triangle is \( \boxed{3} \).

Would you like further details or have any questions? Here are some related questions to consider:

1. How do you find the circumradius of a right triangle?
2. What is Heron's formula for the area of a triangle?
3. How does the inradius relate to the area and semi-perimeter of any triangle?
4. Can the inradius be found for any type of triangle, not just right triangles?
5. What is the relationship between the inradius and the exradii of a triangle?

**Tip:** Recognizing special triangles, like the Pythagorean triplet, can simplify the calculation of the area and other properties.

Learn how to calculate the inradius of a triangle with side lengths 7, 24, 25 using the formula r = A / s. This problem involves recognizing triangle properties, such as the Pythagorean theorem, and applying formulas for the semi-perimeter and area calculation. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation