The height corresponding to the longest side of the triangle whose sides are `42cm, 34cm` and `20 cm` is length is,

Options

A. `15 cm`
B. `36 cm`
C. `16 cm`
D. `23 cm`

Answer

To find the height corresponding to the longest side of the triangle with sides 42 cm, 34 cm, and 20 cm, we can follow these steps:

### Step 1: Identify the longest side
The longest side of the triangle is 42 cm.

### Step 2: Calculate the semi-perimeter (s)
The semi-perimeter \( s \) is calculated using the formula:
\[ s = \frac{a + b + c}{2} \]
where \( a = 42 \, \text{cm} \), \( b = 34 \, \text{cm} \), and \( c = 20 \, \text{cm} \).

Calculating:
\[ s = \frac{42 + 34 + 20}{2} = \frac{96}{2} = 48 \, \text{cm} \]

### Step 3: Use Heron's formula to find the area (A)
Heron's formula states:
\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]
Substituting the values:
\[ A = \sqrt{48(48 - 42)(48 - 34)(48 - 20)} \]
Calculating each term:
\[ A = \sqrt{48 \times 6 \times 14 \times 28} \]

### Step 4: Simplify the area calculation
Calculating the product:
\[ A = \sqrt{48 \times 6 \times 14 \times 28} \]
We can break this down:
\[ 48 = 16 \times 3, \quad 6 = 2 \times 3, \quad 14 = 2 \times 7, \quad 28 = 4 \times 7 \]
Thus,
\[ A = \sqrt{(16 \times 3) \times (2 \times 3) \times (2 \times 7) \times (4 \times 7)} \]
Calculating this gives:
\[ A = \sqrt{48 \times 6 \times 14 \times 28} = \sqrt{336 \times 48} = 336 \, \text{cm}^2 \]

### Step 5: Use the area to find the height (h)
The area can also be expressed as:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
Using the longest side as the base (42 cm):
\[ 336 = \frac{1}{2} \times 42 \times h \]
Solving for \( h \):
\[ 336 = 21h \implies h = \frac{336}{21} = 16 \, \text{cm} \]

### Final Answer
The height corresponding to the longest side of the triangle is \( \boxed{16 \, \text{cm}} \).

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