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

Options

A. `16sqrt5` cm
B. `10sqrt5` cm
C. `24sqrt5` cm
D. 28 cm

Explanation

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

### Step 1: Calculate the semi-perimeter (s) of the triangle.
The semi-perimeter \( s \) is given by the formula:
\[ s = \frac{a + b + c}{2} \]
where \( a = 35 \, \text{cm} \), \( b = 54 \, \text{cm} \), and \( c = 61 \, \text{cm} \).

Calculating:
\[ s = \frac{35 + 54 + 61}{2} = \frac{150}{2} = 75 \, \text{cm} \]

### Step 2: 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:
\[ A = \sqrt{75(75 - 35)(75 - 54)(75 - 61)} \]
Calculating each term:
\[ A = \sqrt{75 \times 40 \times 21 \times 14} \]

### Step 3: Simplify the area calculation.
Calculating the area:
\[ A = \sqrt{75 \times 40 \times 21 \times 14} \]
Breaking it down:
\[ 75 = 3 \times 5^2, \quad 40 = 2^3 \times 5, \quad 21 = 3 \times 7, \quad 14 = 2 \times 7 \]
Combining:
\[ A = \sqrt{(3 \times 5^2) \times (2^3 \times 5) \times (3 \times 7) \times (2 \times 7)} \]
\[ = \sqrt{3^2 \times 5^3 \times 2^4 \times 7^2} \]
Calculating the square root:
\[ A = 3 \times 5^{1.5} \times 2^2 \times 7 = 3 \times 5 \times 2^2 \times 7 \sqrt{5} = 420 \sqrt{5} \, \text{cm}^2 \]

### Step 4: Determine the longest altitude.
The altitude corresponding to the base \( b \) is given by:
\[ h = \frac{2A}{b} \]
To find the longest altitude, we will calculate the altitudes corresponding to each side:

1. **Altitude corresponding to base 35 cm:**
\[ h_a = \frac{2A}{35} = \frac{2 \times 420 \sqrt{5}}{35} = \frac{840 \sqrt{5}}{35} = 24 \sqrt{5} \, \text{cm} \]

2. **Altitude corresponding to base 54 cm:**
\[ h_b = \frac{2A}{54} = \frac{840 \sqrt{5}}{54} = \frac{140 \sqrt{5}}{9} \, \text{cm} \]

3. **Altitude corresponding to base 61 cm:**
\[ h_c = \frac{2A}{61} = \frac{840 \sqrt{5}}{61} \, \text{cm} \]

### Step 5: Compare the altitudes.
To find the longest altitude, we compare \( h_a \), \( h_b \), and \( h_c \). Since \( 35 \) cm is the smallest base, the altitude corresponding to this base will be the longest.

### Final Answer:
The longest altitude of the triangle is:
\[ h_a = 24 \sqrt{5} \, \text{cm} \]

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