In a `DeltaABC` if the length of the sides are `sqrt(2), sqrt(6)` and `sqrt(8)`, then the measures of the angles are

Options

A. `30^(@), 60^(@), 90^(@)`
B. `45^(@), 75^(@), 60^(@)`
C. `45^(@), 30^(@), 105^(@)`
D. none of these

Explanation

To find the measures of the angles in triangle ABC with sides \( a = \sqrt{2} \), \( b = \sqrt{6} \), and \( c = \sqrt{8} \), we will use the Law of Cosines.

### Step 1: Identify the sides
Let:
- \( a = \sqrt{2} \) (opposite angle A)
- \( b = \sqrt{6} \) (opposite angle B)
- \( c = \sqrt{8} \) (opposite angle C)

### Step 2: Use the Law of Cosines to find angle A
The Law of Cosines states:
\[ \cos A = \frac{b^2 + c^2 - a^2}{2bc} \]
Substituting the values:
\[ \cos A = \frac{(\sqrt{6})^2 + (\sqrt{8})^2 - (\sqrt{2})^2}{2 \cdot \sqrt{6} \cdot \sqrt{8}} \]
Calculating the squares:
\[ \cos A = \frac{6 + 8 - 2}{2 \cdot \sqrt{6} \cdot \sqrt{8}} = \frac{12}{2 \cdot \sqrt{48}} = \frac{12}{2 \cdot 4\sqrt{3}} = \frac{12}{8\sqrt{3}} = \frac{3}{2\sqrt{3}} \]
Rationalizing the denominator:
\[ \cos A = \frac{3\sqrt{3}}{6} = \frac{\sqrt{3}}{2} \]
Thus,
\[ A = \cos^{-1}\left(\frac{\sqrt{3}}{2}\right) = 30^\circ \]

### Step 3: Use the Law of Cosines to find angle B
Now we apply the Law of Cosines again to find angle B:
\[ \cos B = \frac{c^2 + a^2 - b^2}{2ac} \]
Substituting the values:
\[ \cos B = \frac{(\sqrt{8})^2 + (\sqrt{2})^2 - (\sqrt{6})^2}{2 \cdot \sqrt{2} \cdot \sqrt{8}} \]
Calculating the squares:
\[ \cos B = \frac{8 + 2 - 6}{2 \cdot \sqrt{2} \cdot \sqrt{8}} = \frac{4}{2 \cdot \sqrt{16}} = \frac{4}{8} = \frac{1}{2} \]
Thus,
\[ B = \cos^{-1}\left(\frac{1}{2}\right) = 60^\circ \]

### Step 4: Find angle C
Using the triangle sum property:
\[ A + B + C = 180^\circ \]
Substituting the known values:
\[ 30^\circ + 60^\circ + C = 180^\circ \]
Solving for C:
\[ C = 180^\circ - 90^\circ = 90^\circ \]

### Conclusion
The measures of the angles in triangle ABC are:
- \( A = 30^\circ \)
- \( B = 60^\circ \)
- \( C = 90^\circ \)

### Final Answer
Thus, the angles of triangle ABC are \( 30^\circ, 60^\circ, \) and \( 90^\circ \).

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