The perimeter of a triangle is 30 cm and its area is `30 cm^(2)`. If the largest side measures 13 cm, what is the lengh of the smallest side of the triangle ?

Options

A. 3 cm
B. 4 cm
C. 5 cm
D. 6 cm

Answer

To find the length of the smallest side of the triangle, we can follow these steps:

### Step 1: Understand the given information
We know:
- The perimeter of the triangle (P) = 30 cm
- The area of the triangle (A) = 30 cm²
- The largest side (c) = 13 cm

### Step 2: Set up the equations
Let the lengths of the sides of the triangle be \( a \), \( b \), and \( c \) (where \( c \) is the largest side). We know:
1. \( a + b + c = 30 \) (Perimeter equation)
2. \( c = 13 \) (Given largest side)

### Step 3: Substitute the value of c into the perimeter equation
Substituting \( c = 13 \) into the perimeter equation:
\[ a + b + 13 = 30 \]
This simplifies to:
\[ a + b = 30 - 13 = 17 \]

### Step 4: Use Heron's formula to find the area
The area of a triangle can also be calculated using Heron's formula:
\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]
where \( s \) is the semi-perimeter:
\[ s = \frac{a + b + c}{2} = \frac{30}{2} = 15 \]

### Step 5: Substitute the values into Heron's formula
Now we can substitute \( s \) and \( c \) into Heron's formula:
\[ 30 = \sqrt{15(15-a)(15-b)(15-13)} \]
This simplifies to:
\[ 30 = \sqrt{15(15-a)(15-b)(2)} \]

### Step 6: Square both sides to eliminate the square root
Squaring both sides gives:
\[ 900 = 15(15-a)(15-b)(2) \]
Dividing both sides by 30:
\[ 30 = (15-a)(15-b) \]

### Step 7: Substitute \( b \) in terms of \( a \)
From \( a + b = 17 \), we can express \( b \) as:
\[ b = 17 - a \]
Substituting this into the equation:
\[ 30 = (15-a)(15-(17-a)) \]
This simplifies to:
\[ 30 = (15-a)(a-2) \]

### Step 8: Expand and rearrange the equation
Expanding the equation:
\[ 30 = 15a - a^2 - 30 + 2a \]
This simplifies to:
\[ a^2 - 17a + 60 = 0 \]

### Step 9: Solve the quadratic equation
Using the quadratic formula \( a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 1, b = -17, c = 60 \):
\[ a = \frac{17 \pm \sqrt{(-17)^2 - 4 \cdot 1 \cdot 60}}{2 \cdot 1} \]
\[ = \frac{17 \pm \sqrt{289 - 240}}{2} \]
\[ = \frac{17 \pm \sqrt{49}}{2} \]
\[ = \frac{17 \pm 7}{2} \]
Calculating the two possible values:
1. \( a = \frac{24}{2} = 12 \)
2. \( a = \frac{10}{2} = 5 \)

### Step 10: Find the smallest side
Since \( a + b = 17 \), if \( a = 12 \), then \( b = 5 \) and vice versa. The smallest side is:
\[ \text{Smallest side} = 5 \text{ cm} \]

### Final Answer
The length of the smallest side of the triangle is **5 cm**.

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