The length of tangent to a circle of radius 2.5 cm from an external point P is 6 cm. Find the distance of P from the nearest point of the circle.

Answer

To find the distance of point P from the nearest point of the circle, we can follow these steps:

### Step 1: Understand the problem
We have a circle with a radius of 2.5 cm and a tangent from an external point P that measures 6 cm. We need to find the distance from point P to the nearest point on the circle.

### Step 2: Draw a diagram
Draw a circle with center O and radius 2.5 cm. Mark point A as the point where the tangent touches the circle. Draw the tangent line from point P to point A, and label the length of the tangent PA as 6 cm.

### Step 3: Identify the triangle
In triangle OAP, we know:
- OA (radius) = 2.5 cm
- PA (length of the tangent) = 6 cm
- OP (hypotenuse) is the distance we want to find.

### Step 4: Apply the Pythagorean theorem
Since OA is perpendicular to PA (the radius is perpendicular to the tangent at the point of contact), we can use the Pythagorean theorem:
\[ OP^2 = OA^2 + PA^2 \]

### Step 5: Substitute the known values
Substituting the known values into the equation:
\[ OP^2 = (2.5)^2 + (6)^2 \]
\[ OP^2 = 6.25 + 36 \]
\[ OP^2 = 42.25 \]

### Step 6: Calculate OP
Now, take the square root to find OP:
\[ OP = \sqrt{42.25} \]
\[ OP = 6.5 \, \text{cm} \]

### Step 7: Find the distance from P to the nearest point on the circle
The distance from point P to the nearest point on the circle (point B) is given by:
\[ PB = OP - OA \]
\[ PB = 6.5 - 2.5 \]
\[ PB = 4 \, \text{cm} \]

### Final Answer
The distance of point P from the nearest point of the circle is **4 cm**.

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