The sides of a triangle are 35 cm, 54 cm and 61 cm respectively. Find the length of its longest altitude.

Options

A. `24 sqrt(2)` cm
B. `24 sqrt(5)` cm
C. `24 sqrt(3)` cm
D. `24 sqrt(7)` cm

Explanation

To find the length of the longest altitude of a triangle with sides 35 cm, 54 cm, and 61 cm, we can follow these steps:

### Step 1: Identify the sides of the triangle
Let the sides of the triangle be:
- \( a = 35 \, \text{cm} \)
- \( b = 54 \, \text{cm} \)
- \( c = 61 \, \text{cm} \)

### Step 2: Calculate the semi-perimeter (s)
The semi-perimeter \( s \) of the triangle is given by the formula:
\[ s = \frac{a + b + c}{2} \]
Substituting the values:
\[ s = \frac{35 + 54 + 61}{2} = \frac{150}{2} = 75 \, \text{cm} \]

### Step 3: Use Heron's formula to find the area (A) of the triangle
Heron's formula states that the area \( A \) can be calculated as:
\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]
Calculating each term:
- \( s - a = 75 - 35 = 40 \)
- \( s - b = 75 - 54 = 21 \)
- \( s - c = 75 - 61 = 14 \)

Now substituting into Heron's formula:
\[ A = \sqrt{75 \times 40 \times 21 \times 14} \]

### Step 4: Calculate the area
Calculating the product:
\[ 75 \times 40 = 3000 \]
\[ 3000 \times 21 = 63000 \]
\[ 63000 \times 14 = 882000 \]
Now, take the square root:
\[ A = \sqrt{882000} \]
To simplify:
\[ A = 30 \sqrt{980} = 30 \times 7 \sqrt{2} = 210 \sqrt{2} \, \text{cm}^2 \]

### Step 5: Find the longest altitude
The altitude corresponding to side \( a \) (35 cm) is the longest altitude. The formula for the area in terms of base and height is:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
Let \( h_a \) be the altitude corresponding to side \( a \):
\[ A = \frac{1}{2} \times a \times h_a \]
Substituting the values:
\[ 210 \sqrt{2} = \frac{1}{2} \times 35 \times h_a \]
Solving for \( h_a \):
\[ 210 \sqrt{2} = 17.5 \times h_a \]
\[ h_a = \frac{210 \sqrt{2}}{17.5} = 12 \sqrt{2} \, \text{cm} \]

### Step 6: Conclusion
Thus, the length of the longest altitude of the triangle is:
\[ h_a = 12 \sqrt{2} \, \text{cm} \]