Two circles touch each other at point X. A common tangent touches them at two distinct points Y and Z. If another tangent through X meets line segment YZ at A and XA = 16 cm, what is the length of YZ (in cm)?

Introduction / Context:
This problem tests a powerful tangent property from geometry: tangents drawn from the same external point to a circle have equal lengths. The setting involves two circles that touch at a point X, which means they share a common tangent line at X. Another common tangent touches the circles at points Y and Z. The tangent at X intersects the common tangent line (YZ) at point A. From point A, there are two tangents to the first circle (AX and AY), and similarly two tangents to the second circle (AX and AZ). Using the equal-tangents-from-an-external-point theorem, we can link AY, AZ, and AX directly. Then, because Y and Z lie on the same line with A between them in this configuration, YZ becomes the sum AY + AZ, which turns into 2*AX. This allows a clean numeric answer without needing radii or angles.

Given Data / Assumptions:

• Two circles touch at X (common tangent at X exists)
• Common external tangent touches circles at Y and Z
• Tangent through X intersects line YZ at A
• XA = 16 cm
• Tangent theorem: from external point A, tangent lengths to a circle are equal

Concept / Approach:
Use equal tangents from A to each circle: For circle 1: AX = AY. For circle 2: AX = AZ. Then YZ = AY + AZ = AX + AX = 2*AX.

Step-by-Step Solution:
From point A, AY is tangent to the first circle at Y and AX is tangent at X So, AY = AX (tangents from same external point to a circle are equal) Similarly, AZ is tangent to the second circle at Z and AX is tangent at X So, AZ = AX Given AX = 16 cm, we get AY = 16 cm and AZ = 16 cm Point A lies on line YZ, so YZ = AY + AZ = 16 + 16 = 32 cm

Verification / Alternative check:
The result does not depend on the radii, only on the tangent equality property. Any valid diagram with the same XA will force AY and AZ to match XA, so the total YZ must be twice XA when A lies between Y and Z on the common tangent line.

Why Other Options Are Wrong:
16 matches only one segment (AY or AZ), not the full YZ. 24 or 18 would imply unequal tangent lengths from A, violating the tangent theorem. 48 incorrectly treats YZ as 3*AX without geometric justification.

Common Pitfalls:
Forgetting that tangents from the same external point are equal, confusing chord lengths with tangent lengths, or adding the wrong segments along line YZ.

Final Answer:
The length of YZ is 32 cm.