From a point outside a circle, a tangent is drawn. The distance from the point to the centre of the circle is 13 centimetres and the largest chord of the circle (its diameter) is 10 centimetres. Calculate the length of the tangent in centimetres.

Introduction / Context:
This question explores the geometry of tangents to a circle from an external point. Tangents are important in many geometric constructions, and the right triangle formed by the radius and the tangent allows us to apply the Pythagoras theorem to find unknown lengths.

Given Data / Assumptions:

• A circle has diameter 10 cm, so radius r = 5 cm.
• Point P is outside the circle.
• Distance OP (from the centre O to point P) = 13 cm.
• PT is the tangent from P to the circle.
• We must find the length PT in centimetres.

Concept / Approach:
A tangent to a circle is perpendicular to the radius at the point of contact. Therefore triangle OPT, where O is the centre and T is the point of tangency, is a right angled triangle with:
OP as the hypotenuse,
OT as the radius,
PT as the tangent length.
We can apply Pythagoras theorem:
OP^2 = OT^2 + PT^2.

Step-by-Step Solution:
Step 1: Compute the radius: diameter = 10 cm, so radius OT = 10 / 2 = 5 cm. Step 2: OP, the distance from the external point to the centre, is given as 13 cm. Step 3: In right triangle OPT, OP^2 = OT^2 + PT^2. Step 4: Substitute values: 13^2 = 5^2 + PT^2. Step 5: Calculate squares: 169 = 25 + PT^2. Step 6: Rearrange: PT^2 = 169 - 25 = 144. Step 7: Take square root: PT = sqrt(144) = 12 cm.

Verification / Alternative check:
We can quickly check if 5, 12, 13 forms a Pythagorean triple. Indeed, 5^2 + 12^2 = 25 + 144 = 169 = 13^2. This confirms that the tangent length 12 cm fits perfectly in the right triangle OPT.

Why Other Options Are Wrong:
8 cm and 10 cm: These do not satisfy the equation 5^2 + PT^2 = 13^2, since 5^2 + 8^2 = 25 + 64 = 89 and 5^2 + 10^2 = 25 + 100 = 125, both not equal to 169.
14 cm and 16 cm: These are too large. For example, 5^2 + 14^2 = 25 + 196 = 221, which is greater than 169, so they cannot be tangent lengths for this configuration.

Common Pitfalls:
Some students mistakenly treat the diameter as the hypotenuse or forget that the tangent is perpendicular to the radius at the point of contact. Others misapply Pythagoras theorem or confuse which side is the hypotenuse. Carefully identify the right angle at the point of tangency and always set the hypotenuse as the side opposite that right angle.

Final Answer:
The length of the tangent is 12 cm.