Find the area of a triangle whose sides are `20 cm, 34 cm` and `42 cm.` Hence find the height corresponding to the longest side.

Options

A. `"16 cm"`
B. `"15 cm"`
C. `"14 cm"`
D. `"13 cm"`

Explanation

To find the area of a triangle with sides of lengths 20 cm, 34 cm, and 42 cm, and then determine the height corresponding to the longest side, follow these steps:

### Step 1: Identify the sides of the triangle
Let the sides of the triangle be:
- \( a = 20 \, \text{cm} \)
- \( b = 34 \, \text{cm} \)
- \( c = 42 \, \text{cm} \)

### Step 2: Calculate the semi-perimeter (s)
The semi-perimeter \( s \) is given by the formula:
\[ s = \frac{a + b + c}{2} \]
Substituting the values:
\[ s = \frac{20 + 34 + 42}{2} = \frac{96}{2} = 48 \, \text{cm} \]

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

Now substituting these values into the formula:
\[ A = \sqrt{48 \times 28 \times 14 \times 6} \]

### Step 4: Calculate the product inside the square root
First, calculate \( 48 \times 28 \):
\[ 48 \times 28 = 1344 \]
Next, calculate \( 1344 \times 14 \):
\[ 1344 \times 14 = 18816 \]
Finally, calculate \( 18816 \times 6 \):
\[ 18816 \times 6 = 112896 \]

### Step 5: Take the square root to find the area
Now, find the square root:
\[ A = \sqrt{112896} = 336 \, \text{cm}^2 \]

### Step 6: Find the height corresponding to the longest side
The longest side \( c = 42 \, \text{cm} \). The area of the triangle can also be expressed as:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
Using the area we calculated:
\[ 336 = \frac{1}{2} \times 42 \times h \]
Solving for \( h \):
\[ 336 = 21h \quad \Rightarrow \quad h = \frac{336}{21} = 16 \, \text{cm} \]

### Final Answer
The area of the triangle is \( 336 \, \text{cm}^2 \) and the height corresponding to the longest side is \( 16 \, \text{cm} \).

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