In a triangle `ABC` if sides a = 13, b =14 and c = 15, then reciprocals of `r_(1),r_(2),r_(3)` are in the ratio

Options

A. `6:7:8`
B. `6:7:8`
C. `8:7:6`
D. none of these

Answer

To solve the problem, we need to find the reciprocals of the inradii \( r_1, r_2, r_3 \) of triangle \( ABC \) with sides \( a = 13 \), \( b = 14 \), and \( c = 15 \), and determine their ratio.

### Step-by-Step Solution:

1. **Calculate the semi-perimeter \( s \)**:
\[ s = \frac{a + b + c}{2} = \frac{13 + 14 + 15}{2} = \frac{42}{2} = 21 \]

**Hint**: The semi-perimeter is half the sum of the triangle's sides.

2. **Calculate the area \( \Delta \) using Heron's formula**:
\[ \Delta = \sqrt{s(s-a)(s-b)(s-c)} \]
First, calculate \( s - a \), \( s - b \), and \( s - c \):
\[ s - a = 21 - 13 = 8, \quad s - b = 21 - 14 = 7, \quad s - c = 21 - 15 = 6 \]
Now substitute these values into Heron's formula:
\[ \Delta = \sqrt{21 \times 8 \times 7 \times 6} \]

**Hint**: Break down the multiplication into smaller parts to simplify calculations.

3. **Calculate \( \Delta \)**:
\[ \Delta = \sqrt{21 \times 8 \times 7 \times 6} = \sqrt{21 \times 336} = \sqrt{7056} \]
To find \( \sqrt{7056} \):
\[ 7056 = 84^2 \quad \Rightarrow \quad \Delta = 84 \]

**Hint**: Use prime factorization or a calculator to find the square root.

4. **Calculate the inradii \( r_1, r_2, r_3 \)**:
\[ r_1 = \frac{\Delta}{s-a} = \frac{84}{8}, \quad r_2 = \frac{\Delta}{s-b} = \frac{84}{7}, \quad r_3 = \frac{\Delta}{s-c} = \frac{84}{6} \]

**Hint**: Remember that \( r_1, r_2, r_3 \) are calculated using the area and the respective segments of the semi-perimeter.

5. **Calculate the reciprocals**:
\[ \frac{1}{r_1} = \frac{8}{84}, \quad \frac{1}{r_2} = \frac{7}{84}, \quad \frac{1}{r_3} = \frac{6}{84} \]

**Hint**: To find the ratio, you can ignore the common denominator.

6. **Determine the ratio of the reciprocals**:
\[ \frac{1}{r_1} : \frac{1}{r_2} : \frac{1}{r_3} = 8 : 7 : 6 \]

**Hint**: Write the ratios in the simplest form by canceling out the common factors.

### Final Answer:
The reciprocals of \( r_1, r_2, r_3 \) are in the ratio \( 8 : 7 : 6 \).