the figure QR is a common tangentogiven circles which meets the tangentTP at P. If QP = 3.8 cm, the length o1QR is:

1. Break Down the Problem

We need to find the length of the common tangent QR given that the length from Q to P (QP) is 3.8 cm.

2. Relevant Concepts

In tangent-secant problems involving circles, the length of the tangent segment from a point outside the circle to the point of tangency can be described using the following relationship: $QR^2 = QP^2 + OP^2$ where O is the center of the circle and OP is the radius.

However, since we are interested in the length of the tangent QR itself, we can use the direct segment concept, where the length from P to the point of contact R on circle Q will also reflect the tangent property.

3. Analysis and Detail

For tangent length, we have: $QR = QP$ since the lengths of tangents from a point outside the circle to the points of tangency (P and R) are equal.

From the problem, we know: $QP = 3.8 \, \text{cm}$

4. Verify and Summarize

Thus, since QR and QP are equal: $QR = QP = 3.8 \, \text{cm}$

Final Answer

The length of QR is $3.8 \, \text{cm}$.