Find the area of the triangle whose sides are 50 cm , 48 cm and 14 cm .Find the heigth of triangle corresponding to the side measuring 48 cm .

Answer

To solve the problem of finding the area of a triangle with sides measuring 50 cm, 48 cm, and 14 cm, and then finding the height corresponding to the side measuring 48 cm, we can follow these steps:

### Step 1: Calculate the semi-perimeter (s) of the triangle.
The semi-perimeter \( s \) is calculated using the formula:
\[ s = \frac{a + b + c}{2} \]
where \( a = 50 \, \text{cm} \), \( b = 48 \, \text{cm} \), and \( c = 14 \, \text{cm} \).

Calculating:
\[ s = \frac{50 + 48 + 14}{2} = \frac{112}{2} = 56 \, \text{cm} \]

### Step 2: Use Heron's formula to find the area of the triangle.
Heron's formula states that the area \( A \) of the triangle can be calculated as:
\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]

Substituting the values:
\[ A = \sqrt{56(56-50)(56-48)(56-14)} \]
Calculating each term:
\[ A = \sqrt{56 \times 6 \times 8 \times 42} \]

### Step 3: Simplify the expression under the square root.
Calculating:
\[ A = \sqrt{56 \times 6 \times 8 \times 42} \]
We can break this down:
\[ 56 = 7 \times 8, \quad 6 = 3 \times 2, \quad 42 = 6 \times 7 \]
So,
\[ A = \sqrt{(7 \times 8) \times (3 \times 2) \times (8) \times (6 \times 7)} \]
Grouping the pairs:
\[ A = \sqrt{(7^2) \times (8^2) \times (3 \times 2) \times 6} \]
Taking the square root:
\[ A = 7 \times 8 \times \sqrt{(3 \times 2) \times 6} = 7 \times 8 \times 6 = 336 \, \text{cm}^2 \]

### Step 4: Find the height corresponding to the base of 48 cm.
To find the height \( h \) corresponding to the base \( b = 48 \, \text{cm} \), we use the area formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
Substituting the known values:
\[ 336 = \frac{1}{2} \times 48 \times h \]

### Step 5: Solve for height \( h \).
Multiplying both sides by 2:
\[ 672 = 48 \times h \]
Now, divide by 48:
\[ h = \frac{672}{48} = 14 \, \text{cm} \]

### Final Answers:
- The area of the triangle is \( 336 \, \text{cm}^2 \).
- The height corresponding to the side measuring \( 48 \, \text{cm} \) is \( 14 \, \text{cm} \).