Multiple choice

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

  1. 18

  2. 24

  3. 16

  4. 32
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D Correct answer
Explanation

For two circles touching at X with a common tangent YZ, the tangent at X bisects YZ. Thus, YZ = 2 * XA. Given XA = 16, YZ = 32.

AI explanation

Using the tangent segments from an external point theorem, the lengths of the two tangents drawn from A to the respective circles are equal, meaning YX and ZX are both equal to AX, which is 16 cm. This property arises because the power of point A with respect to both circles is the same at the point of tangency X. Therefore, the total length of the common tangent YZ is the sum of YX and ZX. Adding the two segments gives 16 cm plus 16 cm, which equals 32 cm.

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