The perimeter of a triangle is 40cm and its area is 60 cm? If the largest side measures 17cm, then the length (in cm) of the smallest side of the triangle is

Options

A. 4
B. 6
C. 8
D. 15

Answer

To solve the problem, we need to find the smallest side of a triangle given the perimeter, area, and the largest side. Here’s a step-by-step solution:

### Step 1: Understand the Given Information
We have:
- Perimeter of the triangle (P) = 40 cm
- Area of the triangle (A) = 60 cm²
- Largest side (C) = 17 cm

### Step 2: Set Up the Equation for the Sides
Let the sides of the triangle be \( A \), \( B \), and \( C \) where \( C \) is the largest side. We know:
\[ A + B + C = 40 \]
Substituting \( C = 17 \):
\[ A + B + 17 = 40 \]
This simplifies to:
\[ A + B = 23 \] (Equation 1)

### Step 3: Use the Area Formula
The area of a triangle can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
To use this, we can consider \( A \) and \( B \) as the base and height. Thus:
\[ 60 = \frac{1}{2} \times A \times B \]
This simplifies to:
\[ A \times B = 120 \] (Equation 2)

### Step 4: Solve the System of Equations
Now we have two equations:
1. \( A + B = 23 \)
2. \( A \times B = 120 \)

From Equation 1, we can express \( B \) in terms of \( A \):
\[ B = 23 - A \]

Substituting this into Equation 2:
\[ A \times (23 - A) = 120 \]
This expands to:
\[ 23A - A^2 = 120 \]
Rearranging gives us a quadratic equation:
\[ A^2 - 23A + 120 = 0 \]

### Step 5: Factor the Quadratic Equation
To solve the quadratic equation, we can factor it:
\[ (A - 15)(A - 8) = 0 \]
This gives us two possible solutions for \( A \):
\[ A = 15 \quad \text{or} \quad A = 8 \]

### Step 6: Find the Corresponding Value of B
Using \( A + B = 23 \):
- If \( A = 15 \), then \( B = 23 - 15 = 8 \)
- If \( A = 8 \), then \( B = 23 - 8 = 15 \)

### Step 7: Identify the Smallest Side
The sides of the triangle are \( 8 \), \( 15 \), and \( 17 \). The smallest side is:
\[ \text{Smallest side} = 8 \text{ cm} \]

### Conclusion
The length of the smallest side of the triangle is **8 cm**.