What is the area of a triangle whose sides are 12 cm, 5 cm and 13 cm?

Options

A. a. `30" cm"^2`
B. b. `60" cm"^2`
C. c. `20 sqrt(3)" cm"^2`
D. d. `20 sqrt(6)" cm"^2`

Answer

To find the area of a triangle with sides measuring 12 cm, 5 cm, and 13 cm, we can follow these steps:

### Step 1: Identify the Type of Triangle
First, we need to determine if the triangle is a right triangle. We can do this using the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.

- Here, the sides are 12 cm, 5 cm, and 13 cm. The longest side is 13 cm.

### Step 2: Apply the Pythagorean Theorem
We check if the triangle satisfies the Pythagorean theorem:
\[ c^2 = a^2 + b^2 \]
where \( c \) is the hypotenuse (13 cm), and \( a \) and \( b \) are the other two sides (12 cm and 5 cm).

Calculating:
\[ 13^2 = 169 \]
\[ 12^2 + 5^2 = 144 + 25 = 169 \]

Since both sides are equal, the triangle is indeed a right triangle.

### Step 3: Use the Area Formula for a Right Triangle
The area \( A \) of a right triangle can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
In this case, we can take the base as 5 cm and the height as 12 cm.

### Step 4: Substitute the Values into the Area Formula
Substituting the values into the formula:
\[ A = \frac{1}{2} \times 5 \times 12 \]

### Step 5: Calculate the Area
Now, we calculate:
\[ A = \frac{1}{2} \times 5 \times 12 = \frac{60}{2} = 30 \text{ cm}^2 \]

### Conclusion
The area of the triangle is \( 30 \text{ cm}^2 \).

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