Two tangent circles and common tangents: Two circles touch at X. A common external tangent touches them at Y and Z. The tangent through X meets YZ at A, and XA = 16 cm. Find YZ (in cm).

Introduction / Context:This configuration uses tangent properties from a common external point. From any external point, the lengths of tangents drawn to a given circle are equal. Here A is a common external point to both circles, with one tangent line AX through X (touching both circles at X) and another common external tangent touching at Y and Z.

Given Data / Assumptions:

• AX is tangent to both circles at X; XA = 16 cm.
• From point A, AY is tangent to the first circle and AZ to the second.

Concept / Approach:For a fixed circle and point A, all tangent lengths from A are equal. Hence, for the first circle, AY = AX = 16, and for the second circle, AZ = AX = 16. The segment YZ along the common external tangent is AY + AZ when measured between tangency points, giving YZ = 16 + 16 = 32 cm.

Step-by-Step Solution:

Verification / Alternative check:Power-of-a-point consistency at A confirms equal tangent lengths to each circle.

Why Other Options Are Wrong:18, 24, 16 undercount one or both tangent lengths.

Common Pitfalls:Thinking YZ equals |AY − AZ|; here both are equal to AX.

Final Answer:32