PQR is an equllateral triangle whose side is 10 cm. What is the value (in cm.) of the inradius of triangle PQR?

Options

A. `5/(sqrt3)`
B. `10sqrt(3)`
C. `10/(sqrt3)`
D. `5sqrt(2)`

Answer

To find the inradius of an equilateral triangle PQR with a side length of 10 cm, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the side length of the triangle**:
The side length (S) of triangle PQR is given as 10 cm.

2. **Use the formula for the inradius of an equilateral triangle**:
The formula for the inradius (r) of an equilateral triangle is:
\[ r = \frac{S}{2\sqrt{3}} \]
where S is the length of the side of the triangle.

3. **Substitute the value of S into the formula**:
Substitute S = 10 cm into the formula:
\[ r = \frac{10}{2\sqrt{3}} \]

4. **Simplify the expression**:
Calculate the value:
\[ r = \frac{10}{2\sqrt{3}} = \frac{10}{2} \cdot \frac{1}{\sqrt{3}} = 5 \cdot \frac{1}{\sqrt{3}} = \frac{5}{\sqrt{3}} \]

5. **Rationalize the denominator** (optional):
To express the inradius in a more standard form, we can rationalize the denominator:
\[ r = \frac{5}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{5\sqrt{3}}{3} \]

6. **Final answer**:
The value of the inradius of triangle PQR is:
\[ r = \frac{5\sqrt{3}}{3} \text{ cm} \]