If the sides of a triangle are in the ratio `1 : sqrt(3) : 2,` then its angles are in the ratio

Options

A. `1:3:5`
B. `2:3:4`
C. `3:2:1`
D. `1:2:3`

Answer

To solve the problem of finding the ratio of angles in a triangle whose sides are in the ratio \(1 : \sqrt{3} : 2\), we can follow these steps:

### Step 1: Assign lengths to the sides
Let the sides of the triangle be:
- \(A = x\)
- \(B = \sqrt{3}x\)
- \(C = 2x\)

### Step 2: Use the Cosine Rule to find angle C
The Cosine Rule states that for any triangle with sides \(a\), \(b\), and \(c\), and opposite angles \(A\), \(B\), and \(C\):
\[ C^2 = A^2 + B^2 - 2AB \cos(C) \]
For our triangle, we can rearrange this to find \(\cos(C)\):
\[ \cos(C) = \frac{A^2 + B^2 - C^2}{2AB} \]

Substituting the values of \(A\), \(B\), and \(C\):
\[ \cos(C) = \frac{x^2 + (\sqrt{3}x)^2 - (2x)^2}{2 \cdot x \cdot \sqrt{3}x} \]
Calculating the squares:
\[ = \frac{x^2 + 3x^2 - 4x^2}{2\sqrt{3}x^2} = \frac{0}{2\sqrt{3}x^2} = 0 \]
Thus, \(\cos(C) = 0\), which means:
\[ C = \cos^{-1}(0) = 90^\circ \]

### Step 3: Use the Cosine Rule to find angle B
Next, we find angle \(B\) using the same rule:
\[ \cos(B) = \frac{A^2 + C^2 - B^2}{2AC} \]
Substituting the values:
\[ \cos(B) = \frac{x^2 + (2x)^2 - (\sqrt{3}x)^2}{2 \cdot x \cdot 2x} \]
Calculating:
\[ = \frac{x^2 + 4x^2 - 3x^2}{4x^2} = \frac{2x^2}{4x^2} = \frac{1}{2} \]
Thus, \(\cos(B) = \frac{1}{2}\), which means:
\[ B = \cos^{-1}\left(\frac{1}{2}\right) = 60^\circ \]

### Step 4: Find angle A
Using the fact that the sum of angles in a triangle is \(180^\circ\):
\[ A + B + C = 180^\circ \]
Substituting the known angles:
\[ A + 60^\circ + 90^\circ = 180^\circ \]
Thus:
\[ A = 180^\circ - 150^\circ = 30^\circ \]

### Step 5: Write the ratio of angles
Now we have:
- \(A = 30^\circ\)
- \(B = 60^\circ\)
- \(C = 90^\circ\)

The ratio of angles \(A : B : C\) is:
\[ 30 : 60 : 90 \]
This simplifies to:
\[ 1 : 2 : 3 \]

### Final Answer
The angles of the triangle are in the ratio \(1 : 2 : 3\).

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