The perimeter of a triangle is 40 cm and its area is `60 cm^(2)`. If the largest side measures 17 cm, 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 length of the smallest side of a triangle given the perimeter, area, and the length of the largest side.

### Step-by-Step Solution:

1. **Identify the given values:**
- Perimeter of the triangle (P) = 40 cm
- Area of the triangle (A) = 60 cm²
- Length of the largest side (c) = 17 cm

2. **Let the lengths of the other two sides be a and b.**
- According to the perimeter, we have:
\[ a + b + c = 40 \]
Substituting the value of c:
\[ a + b + 17 = 40 \]
Simplifying this gives:
\[ a + b = 23 \quad \text{(Equation 1)} \]

3. **Use the area formula for a triangle:**
- The area of a triangle can also be expressed using Heron's formula:
\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]
where \( s \) is the semi-perimeter:
\[ s = \frac{P}{2} = \frac{40}{2} = 20 \]
- Therefore, we can write:
\[ A = \sqrt{20(20-a)(20-b)(20-17)} \]
Simplifying gives:
\[ A = \sqrt{20(20-a)(20-b)(3)} \]
Setting this equal to the area:
\[ 60 = \sqrt{60(20-a)(20-b)} \]
Squaring both sides:
\[ 3600 = 60(20-a)(20-b) \]
Dividing both sides by 60:
\[ 60 = (20-a)(20-b) \quad \text{(Equation 2)} \]

4. **Substituting from Equation 1 into Equation 2:**
- We know \( b = 23 - a \). Substitute this into Equation 2:
\[ 60 = (20-a)(20-(23-a)) \]
Simplifying gives:
\[ 60 = (20-a)(a-3) \]
Expanding this:
\[ 60 = 20a - a^2 - 60 + 3a \]
Rearranging gives:
\[ a^2 - 23a + 120 = 0 \]

5. **Solving the quadratic equation:**
- We can use the quadratic formula:
\[ a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1, b = -23, c = 120 \):
\[ a = \frac{23 \pm \sqrt{(-23)^2 - 4 \cdot 1 \cdot 120}}{2 \cdot 1} \]
\[ a = \frac{23 \pm \sqrt{529 - 480}}{2} \]
\[ a = \frac{23 \pm \sqrt{49}}{2} \]
\[ a = \frac{23 \pm 7}{2} \]
This gives us two possible values for \( a \):
\[ a = \frac{30}{2} = 15 \quad \text{or} \quad a = \frac{16}{2} = 8 \]

6. **Finding the smallest side:**
- If \( a = 15 \), then \( b = 23 - 15 = 8 \).
- If \( a = 8 \), then \( b = 23 - 8 = 15 \).
- Therefore, the smallest side is \( 8 \) cm.

### Final Answer:
The length of the smallest side of the triangle is **8 cm**.