The perimeter of a triangle is 67 cm. The first side is twice the length of the second side.The third is 11 cm more than the second side. Find the length of the shortest side of the triangle.

Options

A. 12 cm
B. 14 cm
C. 17 cm
D. 25 cm

Answer

To solve the problem step by step, we will define the sides of the triangle based on the information given and then use the perimeter to find the lengths of the sides.

### Step 1: Define the sides of the triangle
Let the length of the second side be \( x \) cm.

- The first side, which is twice the length of the second side, will be \( 2x \) cm.
- The third side, which is 11 cm more than the second side, will be \( x + 11 \) cm.

### 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 67 cm. Therefore, we can write the equation as:

\[ x + 2x + (x + 11) = 67 \]

### Step 3: Simplify the equation
Now, combine like terms in the equation:

\[ x + 2x + x + 11 = 67 \]
\[ 4x + 11 = 67 \]

### Step 4: Solve for \( x \)
To isolate \( x \), we first subtract 11 from both sides:

\[ 4x = 67 - 11 \]
\[ 4x = 56 \]

Next, divide both sides by 4:

\[ x = \frac{56}{4} \]
\[ x = 14 \]

### Step 5: Identify the shortest side
Now that we have found \( x = 14 \) cm, we can identify the lengths of all sides:
- Second side = \( x = 14 \) cm
- First side = \( 2x = 2 \times 14 = 28 \) cm
- Third side = \( x + 11 = 14 + 11 = 25 \) cm

The shortest side of the triangle is the second side, which is \( 14 \) cm.

### Final Answer
The length of the shortest side of the triangle is **14 cm**.

---