In a triangle the length of the two larger sides are 24 and 22, respectively.If the angles are in A.P.,then the third side ,is

Options

A. `12+2sqrt(13)`
B. `12-sqrt(13)`
C. `2sqrt(13)+2`
D. `2sqrt(13)-2`

Answer

To find the length of the third side of the triangle given that the two larger sides are 24 and 22, and the angles are in arithmetic progression (A.P.), we can follow these steps:

### Step 1: Define the angles
Let the angles of the triangle be \( A \), \( B \), and \( C \). Since the angles are in A.P., we can express them as:
- \( A = B - d \)
- \( B = B \)
- \( C = B + d \)

Where \( d \) is the common difference.

### Step 2: Use the angle sum property
We know that the sum of angles in a triangle is \( 180^\circ \):
\[ A + B + C = 180^\circ \]
Substituting the expressions for \( A \) and \( C \):
\[ (B - d) + B + (B + d) = 180^\circ \]
This simplifies to:
\[ 3B = 180^\circ \implies B = 60^\circ \]

### Step 3: Find angles A and C
Now that we have \( B \):
\[ A + C = 120^\circ \]
Let \( A = 60 - d \) and \( C = 60 + d \). Thus:
\[ (60 - d) + (60 + d) = 120^\circ \]
This confirms our angles:
- \( A = 60 - d \)
- \( B = 60^\circ \)
- \( C = 60 + d \)

### Step 4: Apply the Law of Cosines
Using the Law of Cosines to find the third side \( C \):
\[ \cos B = \frac{A^2 + C^2 - B^2}{2AC} \]
Substituting \( A = 24 \), \( B = 22 \), and \( B = 60^\circ \):
\[ \cos 60^\circ = \frac{24^2 + C^2 - 22^2}{2 \cdot 24 \cdot C} \]
Since \( \cos 60^\circ = \frac{1}{2} \):
\[ \frac{1}{2} = \frac{576 + C^2 - 484}{48C} \]
This simplifies to:
\[ \frac{1}{2} = \frac{92 + C^2}{48C} \]

### Step 5: Cross-multiply and simplify
Cross-multiplying gives:
\[ 48C = 2(92 + C^2) \]
\[ 48C = 184 + 2C^2 \]
Rearranging this gives:
\[ 2C^2 - 48C + 184 = 0 \]

### Step 6: Solve the quadratic equation
Dividing the entire equation by 2:
\[ C^2 - 24C + 92 = 0 \]
Using the quadratic formula \( C = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ C = \frac{24 \pm \sqrt{(-24)^2 - 4 \cdot 1 \cdot 92}}{2 \cdot 1} \]
\[ C = \frac{24 \pm \sqrt{576 - 368}}{2} \]
\[ C = \frac{24 \pm \sqrt{208}}{2} \]
\[ C = \frac{24 \pm 4\sqrt{13}}{2} \]
\[ C = 12 \pm 2\sqrt{13} \]

### Conclusion
Thus, the length of the third side \( C \) can be either \( 12 + 2\sqrt{13} \) or \( 12 - 2\sqrt{13} \).