If AB =12cm, BC=16 cm and AB is perpendicular to BC, then the radius of the circle passing through the points A, B and C is

Options

A. 6 cm
B. 8 cm
C. 10 cm
D. 12 cm

Answer

To find the radius of the circle passing through points A, B, and C, we can follow these steps:

### Step 1: Understand the Triangle
Given that AB = 12 cm, BC = 16 cm, and AB is perpendicular to BC, we can identify triangle ABC as a right triangle with the right angle at B.

### Step 2: Apply Pythagorean Theorem
In a right triangle, the Pythagorean theorem states that:
\[ AC^2 = AB^2 + BC^2 \]
Substituting the given values:
\[ AC^2 = 12^2 + 16^2 \]

### Step 3: Calculate the Squares
Calculate \( 12^2 \) and \( 16^2 \):
- \( 12^2 = 144 \)
- \( 16^2 = 256 \)

### Step 4: Sum the Squares
Now, add these two results:
\[ AC^2 = 144 + 256 = 400 \]

### Step 5: Find AC
To find AC, take the square root of 400:
\[ AC = \sqrt{400} = 20 \, \text{cm} \]

### Step 6: Determine the Diameter and Radius
Since AC is the diameter of the circle, we can find the radius by dividing the diameter by 2:
\[ \text{Radius} = \frac{AC}{2} = \frac{20}{2} = 10 \, \text{cm} \]

### Final Answer
Thus, the radius of the circle passing through points A, B, and C is **10 cm**.

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