The lengths of the three sides of a triangle are 30 cm, 24 cm and 18 cm respectively. The length of the altitude of the triangle correspondihng to the smallest side is

Options

A. 24 cm
B. 18 cm
C. 30 cm
D. 12 cm

Answer

To find the length of the altitude of the triangle corresponding to the smallest side (which is 18 cm), we can follow these steps:

### Step 1: Identify the sides of the triangle
The sides of the triangle are given as:
- \( a = 30 \) cm
- \( b = 24 \) cm
- \( c = 18 \) cm (smallest side)

### Step 2: Calculate the semi-perimeter (s) of the triangle
The semi-perimeter \( s \) is calculated using the formula:
\[ s = \frac{a + b + c}{2} \]
Substituting the values:
\[ s = \frac{30 + 24 + 18}{2} = \frac{72}{2} = 36 \text{ cm} \]

### Step 3: Calculate the area (A) of the triangle using Heron's formula
Heron's formula states:
\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]
Substituting the values:
\[ A = \sqrt{36(36-30)(36-24)(36-18)} \]
Calculating each term:
\[ A = \sqrt{36 \times 6 \times 12 \times 18} \]
Calculating inside the square root:
\[ A = \sqrt{36 \times 6 = 216} \]
\[ 216 \times 12 = 2592 \]
\[ 2592 \times 18 = 46656 \]
So,
\[ A = \sqrt{46656} = 216 \text{ cm}^2 \]

### Step 4: Calculate the altitude (h) corresponding to the smallest side
The altitude \( h \) corresponding to side \( c \) can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, the base is \( c = 18 \) cm, and we need to find \( h \):
\[ 216 = \frac{1}{2} \times 18 \times h \]
Multiplying both sides by 2:
\[ 432 = 18h \]
Now, divide by 18:
\[ h = \frac{432}{18} = 24 \text{ cm} \]

### Final Answer
The length of the altitude of the triangle corresponding to the smallest side (18 cm) is **24 cm**.

---