In `DeltaPQR`, length of the side QR is less than twice the length of the side PQ by 2 cm. Length of the side PR exceeds the length of the side PQ by 10 cm. The perimeter of 40 cm. The length of the smallest side of `PQR` is:

Options

A. 6 cm
B. 8 cm
C. 7 cm
D. 10 cm

Answer

To solve the problem step by step, we will define the lengths of the sides of triangle PQR based on the information provided in the question.

### Step 1: Define the variables
Let the length of side PQ be \( L \) cm.

### Step 2: Express the lengths of the other sides
According to the question:
- The length of side QR is less than twice the length of side PQ by 2 cm. Therefore, we can express QR as:
\[ QR = 2L - 2 \text{ cm} \]
- The length of side PR exceeds the length of side PQ by 10 cm. Therefore, we can express PR as:
\[ PR = L + 10 \text{ cm} \]

### Step 3: Write the equation for the perimeter
The perimeter of triangle PQR is given as 40 cm. The perimeter is the sum of the lengths of all three sides:
\[ PQ + QR + PR = 40 \]
Substituting the expressions for QR and PR:
\[ L + (2L - 2) + (L + 10) = 40 \]

### Step 4: Simplify the equation
Now, combine like terms:
\[ L + 2L - 2 + L + 10 = 40 \]
This simplifies to:
\[ 4L + 8 = 40 \]

### Step 5: Solve for \( L \)
Now, isolate \( L \) by subtracting 8 from both sides:
\[ 4L = 40 - 8 \]
\[ 4L = 32 \]
Now, divide by 4:
\[ L = 8 \text{ cm} \]

### Step 6: Identify the lengths of all sides
Now that we have \( L \), we can find the lengths of QR and PR:
- \( PQ = L = 8 \text{ cm} \)
- \( QR = 2L - 2 = 2(8) - 2 = 16 - 2 = 14 \text{ cm} \)
- \( PR = L + 10 = 8 + 10 = 18 \text{ cm} \)

### Step 7: Determine the smallest side
The lengths of the sides are:
- \( PQ = 8 \text{ cm} \)
- \( QR = 14 \text{ cm} \)
- \( PR = 18 \text{ cm} \)

The smallest side of triangle PQR is \( PQ \), which is \( 8 \text{ cm} \).

### Final Answer
The length of the smallest side of triangle PQR is \( 8 \text{ cm} \).

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