The sides of a triangle are in the ratio 4:3:2 . If the perimeter of the triangle is 792, what is the length of the smallest side ?

Options

A. 176
B. 200
C. 264
D. 352

Answer

To solve the problem step by step, we will follow the reasoning laid out in the video transcript.

### Step 1: Define the sides of the triangle
Given the sides of the triangle are in the ratio of 4:3:2, we can express the lengths of the sides in terms of a variable \( k \):
- First side = \( 4k \)
- Second side = \( 3k \)
- Third side = \( 2k \)

### Step 2: Write the equation for the perimeter
The perimeter of the triangle is the sum of all its sides. According to the problem, the perimeter is given as 792. Therefore, we can write the equation:
\[ 4k + 3k + 2k = 792 \]

### Step 3: Simplify the equation
Combine the terms on the left side:
\[ (4k + 3k + 2k) = 9k \]
So, we have:
\[ 9k = 792 \]

### Step 4: Solve for \( k \)
To find the value of \( k \), divide both sides of the equation by 9:
\[ k = \frac{792}{9} \]
Calculating this gives:
\[ k = 88 \]

### Step 5: Find the lengths of the sides
Now that we have \( k \), we can find the lengths of all sides:
- First side = \( 4k = 4 \times 88 = 352 \)
- Second side = \( 3k = 3 \times 88 = 264 \)
- Third side = \( 2k = 2 \times 88 = 176 \)

### Step 6: Identify the smallest side
From the calculated lengths:
- First side = 352
- Second side = 264
- Third side = 176

The smallest side is \( 176 \).

### Final Answer
The length of the smallest side of the triangle is \( \boxed{176} \).

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