Find the length of the the tangent from the external point P at a distance of 20 cm from the centre of a circle of radius 12 cm.

Answer

To find the length of the tangent from the external point P to the circle, we can use the Pythagorean theorem. Here’s a step-by-step solution:

### Step 1: Identify the given values
- Distance from the center of the circle (O) to the external point (P): \( OP = 20 \) cm
- Radius of the circle (OA): \( OA = 12 \) cm

### Step 2: Understand the relationship in the triangle
When a tangent is drawn from an external point to a circle, it forms a right triangle with the radius and the line connecting the center of the circle to the external point. In this triangle:
- \( OP \) is the hypotenuse,
- \( OA \) is one leg (the radius),
- \( AP \) is the other leg (the length of the tangent we want to find).

### Step 3: Apply the Pythagorean theorem
According to the Pythagorean theorem:
\[ OP^2 = OA^2 + AP^2 \]
Substituting the known values:
\[ 20^2 = 12^2 + AP^2 \]

### Step 4: Calculate the squares
Calculating the squares:
\[ 400 = 144 + AP^2 \]

### Step 5: Rearrange the equation to solve for \( AP^2 \)
Subtract \( 144 \) from both sides:
\[ AP^2 = 400 - 144 \]
\[ AP^2 = 256 \]

### Step 6: Take the square root to find \( AP \)
Now, take the square root of both sides:
\[ AP = \sqrt{256} \]
\[ AP = 16 \text{ cm} \]

### Final Answer
The length of the tangent from the external point P is \( 16 \) cm.

---