Exercise 10.1: Tangents and Angles of a Circle

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Chapter Overview & SLOs

How do lines that touch the outer perimeter of a circle at a single boundary point establish rigorous right-angled perpendiculars and interconnected angular networks with the center? Tangent geometry transitions circular models into orthogonal computational spaces.

Equivalence of External Tangents: Two tangent lines drawn from a single external point $P$ to a circle at points $Q$ and $R$ have equal lengths: $PQ = PR$.

• Radius perpendicular to tangent: $\angle OQP = 90^\circ$
• Forms right triangle $\triangle OQP$ with hypotenuse $OP$

Pythagorean Radius Computations: Given $OP = 13$ cm (center to external point), $PQ = 12$ cm (tangent length):

• $OP^2 = OQ^2 + PQ^2$ → $13^2 = r^2 + 12^2$
• $169 = r^2 + 144$ → $r^2 = 25$ → $r = 5$ cm

Cyclic Angular Tracking Networks: For tangent-internal triangle systems:

• Supplementary line pairs: $\angle DBC = 180^\circ - \angle PBD$
• Isosceles radial balance: $OD = OB$ (radii) → base angles equal
• Triangle angle sum: $x = 180^\circ - (\text{base angles})$
• Supplementary center angles: $y = 180^\circ - x$

Key Properties:

• Tangent $\perp$ radius at point of contact
• Tangents from external point are equal
• Hypotenuse is $OP$ (center to external point)

These notes are strictly aligned with the Student Learning Outcomes (SLOs) for the FBISE 2026 annual examination.

• How do we apply the geometric property that a tangent line is strictly perpendicular to the radius at its point of contact?
• How do we prove and apply the theorem that the lengths of two tangents drawn from an external point to a circle are equal?
• How do we use the Pythagorean theorem to calculate unknown radii and tangent lengths in right-angled geometric systems?
• How do we track unknown angles across complex circular networks by combining supplementary line pairs with isosceles radial properties?

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