The sides of a triangle are 42 cm, 34 cm and 20 cm. Calculate its area and the length of the height on the longest side. 

Answer

To find the area of the triangle and the height corresponding to the longest side, we can follow these steps:

### Step 1: Identify the sides of the triangle
The sides of the triangle are given as:
- Side A = 42 cm (longest side)
- Side B = 34 cm
- Side C = 20 cm

### Step 2: Calculate the semi-perimeter (s)
The semi-perimeter \( s \) is calculated using the formula:
\[ s = \frac{A + B + C}{2} \]
Substituting the values:
\[ s = \frac{42 + 34 + 20}{2} = \frac{96}{2} = 48 \text{ cm} \]

### Step 3: Apply Heron's formula to find the area
Heron's formula for the area \( A \) of the triangle is:
\[ A = \sqrt{s(s - A)(s - B)(s - C)} \]
Substituting the values:
\[ A = \sqrt{48(48 - 42)(48 - 34)(48 - 20)} \]
Calculating each term:
- \( s - A = 48 - 42 = 6 \)
- \( s - B = 48 - 34 = 14 \)
- \( s - C = 48 - 20 = 28 \)

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

### Step 4: Simplify the area calculation
Calculating the product:
\[ 48 \times 6 = 288 \]
\[ 288 \times 14 = 4032 \]
\[ 4032 \times 28 = 113856 \]
Now taking the square root:
\[ A = \sqrt{113856} \approx 336 \text{ cm}^2 \]

### Step 5: Calculate the height corresponding to the longest side
The height \( h \) corresponding to the base (longest side, which is 42 cm) can be found using the area formula:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
Substituting the known values:
\[ 336 = \frac{1}{2} \times 42 \times h \]
This simplifies to:
\[ 336 = 21h \]
Now, solving for \( h \):
\[ h = \frac{336}{21} = 16 \text{ cm} \]

### Final Results
- Area of the triangle = \( 336 \text{ cm}^2 \)
- Height corresponding to the longest side = \( 16 \text{ cm} \)