What is the area of triangle whose sides are 9cm, 12cm and 15cm?

Options

A. a. 45cm^2
B. b. 54cm^2
C. c. 56cm^2
D. d. 64cm^2

Answer

To find the area of a triangle with sides measuring 9 cm, 12 cm, and 15 cm, we can follow these steps:

### Step 1: Check if the triangle is a right triangle
We will check if the sides 9 cm, 12 cm, and 15 cm form a Pythagorean triplet. For a triangle to be a right triangle, the square of the longest side (hypotenuse) should equal the sum of the squares of the other two sides.

- Let \( a = 9 \), \( b = 12 \), and \( c = 15 \) (where \( c \) is the hypotenuse).
- We will check if \( a^2 + b^2 = c^2 \).

Calculating:
\[ a^2 = 9^2 = 81 \]
\[ b^2 = 12^2 = 144 \]
\[ c^2 = 15^2 = 225 \]

Now, add \( a^2 \) and \( b^2 \):
\[ a^2 + b^2 = 81 + 144 = 225 \]

Since \( a^2 + b^2 = c^2 \) (i.e., \( 225 = 225 \)), the triangle is a right triangle.

### Step 2: Use the formula for the area of a right triangle
The formula for the area \( A \) of a right triangle is:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]

In this case, we can take the base as 9 cm and the height as 12 cm.

### Step 3: Substitute the values into the formula
Substituting the values into the area formula:
\[ A = \frac{1}{2} \times 9 \times 12 \]

### Step 4: Calculate the area
First, calculate \( 9 \times 12 \):
\[ 9 \times 12 = 108 \]

Now, divide by 2:
\[ A = \frac{108}{2} = 54 \]

### Conclusion
The area of the triangle is \( 54 \, \text{cm}^2 \).

---