In a triangle ABC, the internal angle bisector of `/_ABC` meets AC at K. If `BC = 2, CK = 1 and BK =(3sqrt2)/2`, then find the length of side AB.

Answer

To find the length of side AB in triangle ABC given the conditions, we can follow these steps:

### Step 1: Understand the given information
We have:
- \( BC = 2 \)
- \( CK = 1 \)
- \( BK = \frac{3\sqrt{2}}{2} \)

Let \( AK = x \). According to the Angle Bisector Theorem, we have:
\[ \frac{CK}{AK} = \frac{BC}{AB} \]
Substituting the known values:
\[ \frac{1}{x} = \frac{2}{AB} \]

### Step 2: Express AB in terms of x
From the equation above, we can express \( AB \):
\[ AB = 2x \]

### Step 3: Apply the Cosine Rule in triangle BKC
In triangle BKC, we can apply the Cosine Rule:
\[ BK^2 = BC^2 + CK^2 - 2 \cdot BC \cdot CK \cdot \cos(\theta) \]
Substituting the known values:
\[ \left(\frac{3\sqrt{2}}{2}\right)^2 = 2^2 + 1^2 - 2 \cdot 2 \cdot 1 \cdot \cos(\theta) \]
Calculating the squares:
\[ \frac{9 \cdot 2}{4} = 4 + 1 - 4\cos(\theta) \]
\[ \frac{18}{4} = 5 - 4\cos(\theta) \]
\[ \frac{9}{2} = 5 - 4\cos(\theta) \]
Rearranging gives:
\[ 4\cos(\theta) = 5 - \frac{9}{2} \]
\[ 4\cos(\theta) = \frac{10}{2} - \frac{9}{2} = \frac{1}{2} \]
Thus,
\[ \cos(\theta) = \frac{1}{8} \]

### Step 4: Apply the Cosine Rule in triangle ABK
Now, apply the Cosine Rule in triangle ABK:
\[ AB^2 = AK^2 + BK^2 - 2 \cdot AK \cdot BK \cdot \cos(\theta) \]
Substituting \( AB = 2x \), \( AK = x \), and \( BK = \frac{3\sqrt{2}}{2} \):
\[ (2x)^2 = x^2 + \left(\frac{3\sqrt{2}}{2}\right)^2 - 2 \cdot x \cdot \frac{3\sqrt{2}}{2} \cdot \frac{1}{8} \]
Calculating the squares:
\[ 4x^2 = x^2 + \frac{9 \cdot 2}{4} - \frac{3\sqrt{2}}{8}x \]
\[ 4x^2 = x^2 + \frac{9}{4} - \frac{3\sqrt{2}}{8}x \]
Rearranging gives:
\[ 4x^2 - x^2 + \frac{3\sqrt{2}}{8}x - \frac{9}{4} = 0 \]
\[ 3x^2 + \frac{3\sqrt{2}}{8}x - \frac{9}{4} = 0 \]

### Step 5: Solve the quadratic equation
Multiply through by 8 to eliminate the fraction:
\[ 24x^2 + 3\sqrt{2}x - 18 = 0 \]
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ x = \frac{-3\sqrt{2} \pm \sqrt{(3\sqrt{2})^2 - 4 \cdot 24 \cdot (-18)}}{2 \cdot 24} \]
Calculating the discriminant:
\[ (3\sqrt{2})^2 = 18 \]
\[ -4 \cdot 24 \cdot (-18) = 1728 \]
Thus,
\[ x = \frac{-3\sqrt{2} \pm \sqrt{18 + 1728}}{48} \]
\[ x = \frac{-3\sqrt{2} \pm \sqrt{1746}}{48} \]

### Step 6: Find the length of AB
After calculating \( x \), we can find \( AB = 2x \).

### Final Answer
The length of side AB can be calculated from the values obtained for \( x \).

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