The length of tangent from an external point on a circle is always greater than the radius of the circle.

Answer

To solve the question, we need to analyze the statement: "The length of the tangent from an external point on a circle is always greater than the radius of the circle."

### Step-by-Step Solution:

1. **Understanding the Circle and Tangent**:
- Let’s denote the center of the circle as point O and the radius of the circle as r.
- An external point P is located outside the circle.

2. **Drawing the Tangent**:
- From point P, we can draw a tangent line that touches the circle at point A.
- By definition, a tangent to a circle is a line that touches the circle at exactly one point.

3. **Identifying the Radius**:
- The radius OA is the line segment from the center O to the point A where the tangent touches the circle.

4. **Applying the Pythagorean Theorem**:
- In triangle OAP, we can apply the Pythagorean theorem because OA is perpendicular to the tangent PA at point A.
- According to the theorem:
\[ OP^2 = OA^2 + AP^2 \]
- Here, OP is the distance from the external point P to the center O, OA is the radius (r), and AP is the length of the tangent from point P to point A.

5. **Analyzing the Relationship**:
- Rearranging the equation gives:
\[ AP^2 = OP^2 - OA^2 \]
- Since OP is greater than OA (because P is an external point), it follows that:
\[ AP^2 > 0 \quad \text{(because OP > OA)} \]
- Thus, \( AP \) must be greater than 0, but we need to compare it to the radius.

6. **Considering Different Positions of P**:
- If point P is very far from the circle, the length of the tangent PA will be greater than the radius OA.
- If point P is very close to the circle, the length of the tangent PA can approach the radius OA but will never be less than it.

7. **Conclusion**:
- Therefore, the statement "the length of tangent from an external point on a circle is always greater than the radius of the circle" is **false**. The length of the tangent can be equal to or greater than the radius depending on the position of point P.