If the semiperimeter of a right angle triangle isa 126 cm and length of smallest median is 56 cm. Find the area of triangle .

Answer

To find the area of a right-angled triangle given the semi-perimeter and the length of the smallest median, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values**:
- Semi-perimeter (s) = 126 cm
- Length of the smallest median (m) = 56 cm

2. **Understand the Relationship**:
The smallest median in a right triangle can be expressed in terms of the sides of the triangle. The formula for the median (m) from the right angle to the hypotenuse (c) is:
\[ m = \frac{1}{2} \sqrt{2a^2 + 2b^2 - c^2} \]
where \( a \) and \( b \) are the legs of the triangle and \( c \) is the hypotenuse.

3. **Use the Semi-perimeter**:
The semi-perimeter is given by:
\[ s = \frac{a + b + c}{2} \]
From this, we can express the sum of the sides:
\[ a + b + c = 2s = 2 \times 126 = 252 \text{ cm} \]

4. **Express the Area**:
The area (A) of the triangle can also be expressed in terms of the inradius (r) and semi-perimeter (s):
\[ A = r \times s \]

5. **Find the Inradius**:
The inradius (r) can be related to the median:
\[ r = s - 2m \]
Substituting the values we have:
\[ r = 126 - 2 \times 56 = 126 - 112 = 14 \text{ cm} \]

6. **Calculate the Area**:
Now substitute the inradius and semi-perimeter into the area formula:
\[ A = r \times s = 14 \times 126 \]
Calculate:
\[ A = 1764 \text{ cm}^2 \]

### Final Answer:
The area of the triangle is \( 1764 \text{ cm}^2 \).