The sides of a triangle are 45 cm , 60 cm , and 75 cm . Find the length of the altitude drawn to the longest side from its opposite vertex ( in cm) .

Options

A. 27
B. 21
C. 39
D. 36

Answer

To find the length of the altitude drawn to the longest side of the triangle with sides 45 cm, 60 cm, and 75 cm, we will follow these steps:

### Step 1: Identify the sides of the triangle
The sides of the triangle are given as:
- \( a = 45 \, \text{cm} \)
- \( b = 60 \, \text{cm} \)
- \( c = 75 \, \text{cm} \) (the longest side)

### Step 2: Calculate the semi-perimeter (s)
The semi-perimeter \( s \) of the triangle is calculated using the formula:
\[ s = \frac{a + b + c}{2} \]
Substituting the values:
\[ s = \frac{45 + 60 + 75}{2} = \frac{180}{2} = 90 \, \text{cm} \]

### Step 3: Use Heron's formula to find the area of the triangle
Heron's formula for the area \( A \) of the triangle is given by:
\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]
Calculating each term:
- \( s - a = 90 - 45 = 45 \)
- \( s - b = 90 - 60 = 30 \)
- \( s - c = 90 - 75 = 15 \)

Now substituting these values into the area formula:
\[ A = \sqrt{90 \times 45 \times 30 \times 15} \]

### Step 4: Simplify the area calculation
Calculating the product inside the square root:
\[ 90 \times 45 = 4050 \]
\[ 4050 \times 30 = 121500 \]
\[ 121500 \times 15 = 1822500 \]
Now taking the square root:
\[ A = \sqrt{1822500} \]
To simplify \( \sqrt{1822500} \), we can factor it:
\[ 1822500 = 1350^2 \]
Thus,
\[ A = 1350 \, \text{cm}^2 \]

### Step 5: Calculate the altitude (h) to the longest side
The area of a triangle can also be expressed as:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, the base is the longest side \( c = 75 \, \text{cm} \), and the height is \( h \):
\[ 1350 = \frac{1}{2} \times 75 \times h \]
Multiplying both sides by 2:
\[ 2700 = 75h \]
Now, solving for \( h \):
\[ h = \frac{2700}{75} = 36 \, \text{cm} \]

### Final Answer
The length of the altitude drawn to the longest side from its opposite vertex is \( 36 \, \text{cm} \).

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