In a circle geometry problem, PAB and PCD are two secants drawn from an external point P to the same circle. The lengths of the segments are as follows: PA = 10 cm, AB = 12 cm and PC = 11 cm. Using the secant–secant power theorem, what is the length (in cm) of the outer segment PD?

Introduction / Context:
This question is based on circle geometry and specifically on the secant–secant power theorem. Two secants PAB and PCD are drawn from an external point P to a circle. The aim is to find the unknown outer segment PD using the given lengths of other segments. Such questions are very common in aptitude tests and competitive exams and test how well a student remembers and applies the right circle theorems in a numerical situation.

Given Data / Assumptions:

• PAB and PCD are two secants from the same external point P to a circle.
• PA = 10 cm (near segment of the first secant).
• AB = 12 cm, therefore PB = PA + AB = 10 + 12 = 22 cm.
• PC = 11 cm (near segment of the second secant).
• Let PD be the full length of the second secant from P to D.
• The circle is standard and the usual secant–secant theorem applies.

Concept / Approach:
The key theorem is the secant–secant power theorem from circle geometry. It states that if two secants PAB and PCD are drawn from an external point P, then the product of the entire length of one secant and its external segment equals the product of the entire length of the other secant and its external segment. Symbolically, if PA is the external part of the first secant and PB is the far end, and PC, PD are the corresponding parts of the second secant, then:
PA * PB = PC * PD This relation allows us to find the missing length PD once all other lengths are known.

Step-by-Step Solution:
Step 1: Compute PB, the total length of the first secant = PA + AB = 10 + 12 = 22 cm. Step 2: Write the secant–secant power equation: PA * PB = PC * PD. Step 3: Substitute the known values: 10 * 22 = 11 * PD. Step 4: Compute the left side: 10 * 22 = 220. Step 5: Solve for PD: 11 * PD = 220, therefore PD = 220 / 11 = 20 cm.

Verification / Alternative check:
We can verify the result by rechecking the products. For the first secant, PA * PB = 10 * 22 = 220. For the second secant using our answer, PC * PD = 11 * 20 = 220. Since both products match, the secant–secant theorem is satisfied. This confirms that PD = 20 cm is consistent and correct. The value also lies comfortably in the range of given options and is greater than PC, which makes sense for a full secant length measured from the same external point.

Why Other Options Are Wrong:
Option 18 gives PC * PD = 11 * 18 = 198, which does not match 220 and therefore violates the theorem. Option 9 gives 11 * 9 = 99, which is much too small. Option 12 gives 11 * 12 = 132, which again does not equal 220. Option 16 gives 11 * 16 = 176, which is also incorrect. Only PD = 20 gives the correct product 220 on both sides of the equation.

Common Pitfalls:
A common error is to treat PA and AB incorrectly, for example using only AB in the formula instead of the full length PB = PA + AB. Another frequent mistake is to mix up which segment is external and which is total, which leads to incorrect products. Students also sometimes attempt to use Pythagoras theorem even though this is not a right triangle configuration. Remember that the secant–secant theorem applies directly whenever two secants originate from the same external point. Carefully adding PA and AB to get PB before substituting prevents most errors.

Final Answer:
Hence, the correct length of PD is 20 centimetres.