The angles of a triangle are in the ratio `2:3:7` and the radius of the circumscribed circle is 10 cm . The length of the smallest side is

Options

A. 2 m
B. 5 cm
C. 7 cm
D. 10 cm

Answer

To solve the problem step by step, we will follow these instructions:

### Step 1: Determine the Angles of the Triangle
Given the ratio of the angles is \(2:3:7\), we can express the angles in terms of a variable \(x\):
- Let the angles be \(2x\), \(3x\), and \(7x\).

### Step 2: Set Up the Equation for the Sum of Angles
The sum of the angles in a triangle is \(180^\circ\):
\[ 2x + 3x + 7x = 180^\circ \]
This simplifies to:
\[ 12x = 180^\circ \]

### Step 3: Solve for \(x\)
Now, divide both sides by \(12\):
\[ x = \frac{180^\circ}{12} = 15^\circ \]

### Step 4: Calculate the Angles
Now we can find the angles:
- \(A = 2x = 2 \times 15^\circ = 30^\circ\)
- \(B = 3x = 3 \times 15^\circ = 45^\circ\)
- \(C = 7x = 7 \times 15^\circ = 105^\circ\)

### Step 5: Use the Circumradius Formula
The formula for the sides of a triangle in relation to the circumradius \(R\) is:
\[ a = 2R \sin A, \quad b = 2R \sin B, \quad c = 2R \sin C \]
Given \(R = 10\) cm, we can calculate each side.

### Step 6: Calculate Side \(a\)
Using \(A = 30^\circ\):
\[ a = 2 \times 10 \times \sin(30^\circ) = 20 \times \frac{1}{2} = 10 \text{ cm} \]

### Step 7: Calculate Side \(b\)
Using \(B = 45^\circ\):
\[ b = 2 \times 10 \times \sin(45^\circ) = 20 \times \frac{1}{\sqrt{2}} = 10\sqrt{2} \text{ cm} \]

### Step 8: Calculate Side \(c\)
Using \(C = 105^\circ\):
\[ c = 2 \times 10 \times \sin(105^\circ) = 20 \times \sin(105^\circ) \]
Using the sine addition formula:
\[ \sin(105^\circ) = \sin(60^\circ + 45^\circ) = \sin(60^\circ)\cos(45^\circ) + \cos(60^\circ)\sin(45^\circ) \]
Calculating:
\[ = \frac{\sqrt{3}}{2} \cdot \frac{1}{\sqrt{2}} + \frac{1}{2} \cdot \frac{1}{\sqrt{2}} = \frac{\sqrt{3} + 1}{2\sqrt{2}} \]
Thus:
\[ c = 20 \cdot \frac{\sqrt{3} + 1}{2\sqrt{2}} = 10 \cdot \frac{\sqrt{3} + 1}{\sqrt{2}} \text{ cm} \]

### Step 9: Identify the Smallest Side
Now we compare the lengths:
- Side \(a = 10\) cm
- Side \(b = 10\sqrt{2} \approx 14.14\) cm
- Side \(c = 10 \cdot \frac{\sqrt{3} + 1}{\sqrt{2}} \approx 10 \cdot 1.414 \approx 14.14\) cm

The smallest side is \(a\).

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

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