A circle has its centre at O. From a point P outside the circle, a tangent is drawn that touches the circle at A. If PA = 4 centimetres and PO = 5 centimetres, find the length of the radius of the circle in centimetres.

Introduction / Context:
This question is another example of using tangents to a circle and the right triangle formed by the radius, tangent and line from the centre to the external point. It tests your ability to identify the correct right triangle and then apply Pythagoras theorem to determine the radius of the circle using the given distances.

Given Data / Assumptions:

• O is the centre of the circle.
• P is an external point outside the circle.
• PA is a tangent from P to the circle, touching at point A.
• PA = 4 cm.
• PO, the distance from P to the centre O, is 5 cm.
• We must find the radius OA of the circle.
• The radius OA is perpendicular to the tangent PA at point A.

Concept / Approach:
Since the radius drawn to a tangent at the point of contact is perpendicular to the tangent, angle OAP is 90 degrees. Therefore triangle OAP is a right angled triangle with hypotenuse OP, one leg OA (the radius) and the other leg AP (the tangent). Applying Pythagoras theorem:
OP^2 = OA^2 + AP^2.
We substitute the known values and solve for OA, which directly gives the radius.

Step-by-Step Solution:
Step 1: Recognise that triangle OAP is right angled at A because OA is perpendicular to tangent PA. Step 2: OP is the hypotenuse with length 5 cm. Step 3: AP is one leg with length 4 cm. Step 4: Let OA = r be the radius we need to find. Step 5: Apply Pythagoras theorem: OP^2 = OA^2 + AP^2. Step 6: Substitute known values: 5^2 = r^2 + 4^2. Step 7: Compute squares: 25 = r^2 + 16. Step 8: Rearrange: r^2 = 25 − 16 = 9. Step 9: Take square root: r = 3 cm.

Verification / Alternative check:
We can confirm by checking the Pythagorean triple. The lengths 3, 4 and 5 form a classic right triangle. Here, OA = 3 cm, AP = 4 cm and OP = 5 cm. Since 3^2 + 4^2 = 9 + 16 = 25 = 5^2, the right triangle condition is satisfied, validating that the radius of the circle is 3 cm.

Why Other Options Are Wrong:
1 cm and 2 cm: These radii are too small; they would not satisfy 5^2 = r^2 + 4^2 when substituted into Pythagoras theorem.
4 cm: This value would give 5^2 = 4^2 + 4^2 = 32, which is not true.
5 cm: This would imply the tangent length is zero (right triangle degenerates), which contradicts PA = 4 cm.

Common Pitfalls:
Students sometimes confuse which segment is the hypotenuse and may subtract instead of adding in Pythagoras theorem. Others may forget that the tangent and radius are perpendicular at the point of contact, leading them to apply the theorem incorrectly. Drawing the right triangle clearly and marking the right angle at A helps in correctly identifying the sides and performing the calculation.

Final Answer:
The radius of the circle is 3 cm.