On a transmission line, the measured magnitudes of the input impedance at open-circuit (Z_in,OC) and short-circuit (Z_in,SC) are 100 Ω and 25 Ω, respectively. The characteristic impedance Z0 is:

Introduction:
For a (low-loss) transmission line section of unknown electrical length, the open-circuit and short-circuit input impedances can be used to determine the characteristic impedance Z0. This is a common lab technique in RF measurements.

Given Data / Assumptions:

• |Z_in,OC| = 100 Ω
• |Z_in,SC| = 25 Ω
• Assume a low-loss line so that |Z0| ≈ sqrt(|Z_in,OC| * |Z_in,SC|)

Concept / Approach:
For a uniform line, Z0 can be obtained from the geometric mean of OC and SC input impedances: Z0 ≈ sqrt(Z_OC * Z_SC) (magnitude relation under low-loss assumption).

Step-by-Step Solution:

Verification / Alternative check:
Network analyzer methods and line calculators confirm that for practical low-loss coaxial lines, the geometric mean relation holds closely.

Why Other Options Are Wrong:

• A/D: 25 Ω and 100 Ω are the raw OC/SC magnitudes, not Z0.
• C: 75 Ω does not satisfy the geometric-mean relationship.

Common Pitfalls:
Forgetting that this relationship applies to low-loss lines; mixing up arithmetic vs geometric mean.

Final Answer:
50 Ω