The length of the circum-radius of a triangle having sides of length 12cm, 16cm and 20 cm is

Options

A. 15cm
B. 10cm
C. 18cm
D. 16cm

Answer

To find the circumradius of a triangle with sides of lengths 12 cm, 16 cm, and 20 cm, we can follow these steps:

### Step 1: Identify the type of triangle
First, we need to check if the triangle is a right triangle. We can use the Pythagorean theorem for this purpose. The longest side (20 cm) will be considered as the hypotenuse.

### Step 2: Apply the Pythagorean theorem
According to the Pythagorean theorem:
\[ c^2 = a^2 + b^2 \]
where \( c \) is the hypotenuse and \( a \) and \( b \) are the other two sides.

Here, let:
- \( c = 20 \) cm
- \( a = 12 \) cm
- \( b = 16 \) cm

Now, we calculate:
\[ 20^2 = 12^2 + 16^2 \]
\[ 400 = 144 + 256 \]
\[ 400 = 400 \]

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

### Step 3: Use the circumradius formula for a right triangle
For a right triangle, the circumradius \( R \) can be calculated using the formula:
\[ R = \frac{c}{2} \]
where \( c \) is the length of the hypotenuse.

### Step 4: Calculate the circumradius
Now substituting the value of \( c \):
\[ R = \frac{20}{2} = 10 \text{ cm} \]

### Conclusion
The length of the circumradius of the triangle is **10 cm**.

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