From a point P outside a circle with centre O and radius 6 centimetres, a tangent PT is drawn to the circle. If PT = 8 centimetres, find the length of segment OP in centimetres.

Introduction / Context:
This problem involves a tangent to a circle from an external point and uses a classic right triangle configuration. The radius drawn to the point of tangency is perpendicular to the tangent, giving a right triangle. Applying Pythagoras theorem then allows us to find the distance from the external point to the centre of the circle.

Given Data / Assumptions:

• A circle has centre O and radius 6 cm.
• P is a point outside the circle.
• PT is a tangent from P to the circle, touching it at point T.
• The length of PT is 8 cm.
• We must find the length of OP in centimetres.
• The radius OT is perpendicular to tangent PT at the point of contact T.

Concept / Approach:
The key concept is that a radius drawn to a point of tangency is perpendicular to the tangent. This means that triangle OPT is a right angled triangle with right angle at T. The hypotenuse is OP, one leg is OT (the radius) and the other leg is PT (the tangent). Using Pythagoras theorem:
OP^2 = OT^2 + PT^2.
We substitute the known values and solve for OP.

Step-by-Step Solution:
Step 1: Identify the right triangle OPT where angle at T is 90 degrees. Step 2: OT = radius = 6 cm. Step 3: PT = tangent length = 8 cm. Step 4: Apply Pythagoras theorem: OP^2 = OT^2 + PT^2. Step 5: Substitute values: OP^2 = 6^2 + 8^2 = 36 + 64. Step 6: Compute: 36 + 64 = 100. Step 7: Therefore OP^2 = 100, so OP = sqrt(100) = 10 cm.

Verification / Alternative check:
We can recognise the triple (6, 8, 10) as a scaled version of the well known Pythagorean triple (3, 4, 5). The ratio 6 : 8 : 10 simplifies to 3 : 4 : 5, confirming that the sides indeed form a right triangle. This provides an additional intuitive check that OP is correctly calculated as 10 cm.

Why Other Options Are Wrong:
12 cm and 16 cm: These values are too large when compared with the leg lengths 6 and 8. Plugging them back into Pythagoras theorem does not give equality for 6^2 + 8^2.
9 cm and 8 cm: These are too small to be the hypotenuse of a right triangle with legs 6 and 8, since the hypotenuse must be longer than either leg and satisfy OP^2 = 36 + 64.

Common Pitfalls:
Common errors include forgetting that the radius is perpendicular to the tangent, misidentifying the hypotenuse or incorrectly computing the sum of squares. Some students attempt to subtract instead of add when using Pythagoras theorem, which is incorrect for finding the hypotenuse. Always remember that for a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.

Final Answer:
The length of OP is 10 cm.