The radii `r_(1), r_(2), r_(3)` of the escribed circles of the triangle ABC are in H.P. If the area of the triangle is `24 cm^(2)` and its perimeter is 24 cm, then the length of its largest side is

Options

A. 10
B. 9
C. 8
D. none of these

Explanation

To solve the problem step by step, we will follow the reasoning provided in the video transcript.

### Step-by-Step Solution:

1. **Understanding the Given Information:**
We know that the radii \( r_1, r_2, r_3 \) of the escribed circles of triangle \( ABC \) are in Harmonic Progression (H.P.). This implies that:
\[ 2b = a + c \]
where \( a, b, c \) are the lengths of the sides opposite to vertices \( A, B, C \) respectively.

2. **Using the Perimeter:**
The perimeter of the triangle is given as \( 24 \) cm. Therefore:
\[ a + b + c = 24 \]
This can also be expressed in terms of the semi-perimeter \( s \):
\[ 2s = 24 \implies s = 12 \]

3. **Substituting for \( a + c \):**
From the H.P. condition, we have:
\[ a + c = 2b \]
Substituting this into the perimeter equation:
\[ 2b + b = 24 \implies 3b = 24 \implies b = 8 \]

4. **Finding \( a + c \):**
Now substituting \( b \) back into the equation for \( a + c \):
\[ a + c = 2b = 2 \times 8 = 16 \]

5. **Expressing \( c \) in terms of \( a \):**
We can express \( c \) as:
\[ c = 16 - a \]

6. **Using the Area of the Triangle:**
The area \( A \) of the triangle is given as \( 24 \) cm². The formula for the area in terms of the semi-perimeter \( s \) and the sides is:
\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]
Substituting the known values:
\[ 24 = \sqrt{12(12-a)(12-8)(12-c)} \]
Simplifying, we have:
\[ 24 = \sqrt{12(12-a)(4)(12-(16-a))} \]
\[ 24 = \sqrt{12(12-a)(4)(a-4)} \]
\[ 24 = \sqrt{48(12-a)(a-4)} \]

7. **Squaring Both Sides:**
Squaring both sides gives:
\[ 576 = 48(12-a)(a-4) \]
Dividing by \( 48 \):
\[ 12 = (12-a)(a-4) \]

8. **Expanding and Rearranging:**
Expanding the right side:
\[ 12 = 12a - 48 - a^2 + 4a \]
Rearranging gives:
\[ a^2 - 16a + 60 = 0 \]

9. **Factoring the Quadratic Equation:**
Factoring the quadratic:
\[ (a - 6)(a - 10) = 0 \]
Thus, \( a = 6 \) or \( a = 10 \).

10. **Finding the Largest Side:**
Since \( a \) represents one side of the triangle, we find \( c \) for both values:
- If \( a = 6 \), then \( c = 16 - 6 = 10 \).
- If \( a = 10 \), then \( c = 16 - 10 = 6 \).

The sides of the triangle are \( a = 10, b = 8, c = 6 \). The largest side is:
\[ \text{Largest side} = 10 \text{ cm} \]

### Final Answer:
The length of the largest side is \( 10 \) cm.