Find the area of triangle whose sides are 5 cm, 12 cm and 13 cm. Also, find the length of its altitude corresponding to the longest side.

Answer

To find the area of the triangle with sides 5 cm, 12 cm, and 13 cm, and the length of its altitude 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:
- a = 5 cm
- b = 12 cm
- c = 13 cm (the longest 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:
\[ s = \frac{5 + 12 + 13}{2} = \frac{30}{2} = 15 \text{ cm} \]

### Step 3: Apply Heron's formula to find the area
Heron's formula for the area \( A \) of a triangle is given by:
\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]
Substituting the values we found:
\[ A = \sqrt{15(15-5)(15-12)(15-13)} \]
Calculating each term:
- \( s - a = 15 - 5 = 10 \)
- \( s - b = 15 - 12 = 3 \)
- \( s - c = 15 - 13 = 2 \)

Now substituting these values back into the formula:
\[ A = \sqrt{15 \times 10 \times 3 \times 2} \]
Calculating the product:
\[ 15 \times 10 = 150 \]
\[ 150 \times 3 = 450 \]
\[ 450 \times 2 = 900 \]
Now take the square root:
\[ A = \sqrt{900} = 30 \text{ cm}^2 \]

### Step 4: Find the altitude corresponding to the longest side
The area of the triangle can also be expressed using the base and height (altitude):
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, the base is the longest side (13 cm), and we need to find the height (h):
\[ 30 = \frac{1}{2} \times 13 \times h \]
Multiplying both sides by 2:
\[ 60 = 13h \]
Now, solving for \( h \):
\[ h = \frac{60}{13} \approx 4.615 \text{ cm} \]

### Final Answers
- Area of the triangle: \( 30 \text{ cm}^2 \)
- Length of the altitude corresponding to the longest side: \( \approx 4.615 \text{ cm} \)