Write True or False and justify your answer:
In a triangle, the sides are given as `11` `cm`, `12` `cm` and `13` `cm`. The length of the altitude is `10.25` `cm` corresponding to the side having `12` cm.

Explanation

To determine whether the statement is true or false, we will follow these steps:

### Step 1: Identify the sides of the triangle
The sides of the triangle are given as:
- \( a = 11 \, \text{cm} \)
- \( b = 12 \, \text{cm} \)
- \( c = 13 \, \text{cm} \)

### 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:
\[ s = \frac{11 + 12 + 13}{2} = \frac{36}{2} = 18 \, \text{cm} \]

### Step 3: Use Heron's formula to find the area of the triangle
Heron's formula states that the area \( A \) of a triangle can be calculated as:
\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]
Substituting the values:
\[ A = \sqrt{18 \times (18 - 11) \times (18 - 12) \times (18 - 13)} \]
Calculating the terms inside the square root:
\[ A = \sqrt{18 \times 7 \times 6 \times 5} \]
Calculating this step-by-step:
- \( 18 \times 7 = 126 \)
- \( 126 \times 6 = 756 \)
- \( 756 \times 5 = 3780 \)

Now, we compute the square root:
\[ A = \sqrt{3780} \]

### Step 4: Approximate the area
To find the approximate value of \( \sqrt{3780} \):
\[ \sqrt{3780} \approx 61.5 \, \text{cm}^2 \]

### Step 5: Calculate the area using the base and altitude
The area can also be calculated using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, the base is \( 12 \, \text{cm} \) and the height (altitude) is \( 10.25 \, \text{cm} \):
\[ A = \frac{1}{2} \times 12 \times 10.25 \]
Calculating this:
\[ A = 6 \times 10.25 = 61.5 \, \text{cm}^2 \]

### Step 6: Compare the two areas
We have calculated the area using both methods:
- Area from Heron's formula: \( 61.5 \, \text{cm}^2 \)
- Area from base and altitude: \( 61.5 \, \text{cm}^2 \)

Since both areas are equal, we conclude that the statement is **True**.

### Justification
The area calculated using Heron's formula and the area calculated using the base and altitude are equal, confirming that the given altitude of \( 10.25 \, \text{cm} \) corresponding to the side of \( 12 \, \text{cm} \) is correct.

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