Question 5

Proof by Contradiction: A technique where we assume the opposite of what we want to prove, then show that this assumption leads to an impossible situation. Since the assumption cannot be true, the original statement must be true.

Tangent: A line that touches a circle at exactly one point (the point of contact) without crossing through it.

Tangent-Radius Perpendicularity Theorem: The radius drawn to the point of tangency is always perpendicular to the tangent.

We will prove this theorem by contradiction.

Step 1 — Assume the opposite

Let's consider a circle. Let O be its centre. Let AB be a tangent line. It touches the circle at point P. We want to prove a theorem. The perpendicular to AB at P passes through O. Let's assume the opposite of what we want to prove.

Assumption: The perpendicular to AB at P does NOT pass through the centre O. Instead, let it pass through some other point O'. So O'P ⊥ AB.

Assume the perpendicular at P does not pass through O. Let it pass through another point O'. So, O'P is perpendicular to AB.

$\boxed{\angle O'PB = 90^\circ}$

Step 2 — Use the radius property

Tangent-Radius Perpendicularity Theorem: The radius to the point of contact is always perpendicular to the tangent.

We know a property of circles. The radius is perpendicular to the tangent. This happens at the point of contact. So, radius OP is perpendicular to tangent AB.

$\boxed{\angle OPB = 90^\circ}$

Step 3 — Find the contradiction

From Step 1, we found:

$\angle O'PB = 90^\circ$

From Step 2, we found:

$\angle OPB = 90^\circ$

This means that:

$\angle O'PB = \angle OPB$

Both lines O'P and OP start from point P. Both lines are perpendicular to AB. Uniqueness of Perpendicular: Through any given point on a line, only one perpendicular line can be drawn. Since both O'P and OP are perpendicular to AB at the same point P, they must be the same line.

There can only be one line perpendicular to AB through point P. Therefore, lines O'P and OP must be the same line. This means point O' must lie on the line OP. For O'P to be the perpendicular from P, O' must be the centre O. This contradicts our initial assumption that the perpendicular does not pass through O.

Conclusion: Since our assumption leads to a contradiction, the assumption must be false. Therefore, the perpendicular at the point of contact to the tangent must pass through the centre O.

$\boxed{\text{O' coincides with O}}$

Answer

The perpendicular at the point of contact to the tangent to a circle passes through the centre.