Find the area of a triangle whose sides are 28 cm , 21 cm and 35 cm . Also find the length of the altitude corresponding to the largest side of the triangle .

Answer

To find the area of a triangle with sides measuring 28 cm, 21 cm, and 35 cm, and to calculate the length of the altitude corresponding to the largest side, we will follow these steps:

### Step 1: Identify the sides of the triangle
The sides of the triangle are:
- a = 28 cm
- b = 21 cm
- c = 35 cm (largest side)

### 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:
\[ s = \frac{28 + 21 + 35}{2} = \frac{84}{2} = 42 \text{ cm} \]

### Step 3: Use Heron's formula to find the area (A)
Heron's formula for the area of a triangle is given by:
\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]
Substituting the values:
\[ A = \sqrt{42 \times (42 - 28) \times (42 - 21) \times (42 - 35)} \]
Calculating the individual terms:
- \( s - a = 42 - 28 = 14 \)
- \( s - b = 42 - 21 = 21 \)
- \( s - c = 42 - 35 = 7 \)

Now substituting these values back into the area formula:
\[ A = \sqrt{42 \times 14 \times 21 \times 7} \]

### Step 4: Calculate the area
Calculating the product:
\[ 42 \times 14 = 588 \]
\[ 588 \times 21 = 12348 \]
\[ 12348 \times 7 = 86436 \]
Now, take the square root:
\[ A = \sqrt{86436} \approx 294 \text{ cm}^2 \]

### Step 5: Find the altitude corresponding to the largest side (c = 35 cm)
The area can also be expressed using the base and height:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
Using the largest side as the base:
\[ 294 = \frac{1}{2} \times 35 \times h \]
Solving for \( h \):
\[ 294 = 17.5 \times h \]
\[ h = \frac{294}{17.5} \approx 16.8 \text{ cm} \]

### Final Results
- The area of the triangle is \( 294 \text{ cm}^2 \).
- The length of the altitude corresponding to the largest side is \( 16.8 \text{ cm} \).