Tangents to a Circle

11-07-2023 app.cch No Comments

If $P Q$ is a tangent to the circle at $T$ , then $O T$ is perpendicular to $P Q$ .

Condition: $P Q$ is a tangent at $T$

Result: $O T ⊥ P Q$

Reference: tangent $⊥$ radius
If $O T$ is perpendicular to $P T Q$ , then $P Q$ is a tangent to the circle at $T$ .

Condition: $O T ⊥ P T Q$

Result: $P Q$ is a tangent

Reference: converse of tangent $⊥$ radius
If $P Q$ is a tangent to the circle at $T$ , any straight line perpendicular to $P Q$ and passing through $T$ must pass through the centre.

Condition: $P Q$ is a tangent at $T$ and a perpendicular line passing through $T$

Result: The straight line must pass through the centre

Reference: line passing through the pt of contact $⊥$ tangent passes through centre
Given that $P T$ and $Q T$ are two tangents to the same circle. Then $Δ O P T ≅ Δ O Q T$ .

Condition: Two tangents $P T$ and $Q T$

Result:
1. $P T = Q T$
2. $∠ P T O = ∠ Q T O$
3. $∠ T O P = ∠ T O Q$
Reference: tangent properties
If $P Q$ is a tangent to the circle, then the angles in alternate segment equal.

Condition: $P Q$ is a tangent

Result: $∠ A B C = ∠ C A Q$

Reference: $∠$ in alt. segment
If the angles in the alternate segment equal, then $P Q$ is a tangent to the circle at $A$ .

Condition: $∠ A B C = ∠ C A Q$

Result: $P Q$ is a tangent to the circle

Reference: converse of $∠$ in alt. segment

Math, Measures, Shape and Space, Properties of Circle, Revision Note

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