Exercise 11.1: Practical Geometry of Circles

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Chapter Overview & SLOs

How do we translate geometric axioms into precise physical constructions using only an unmarked straightedge and a compass, uncovering hidden central centers and drawing perfect tangent vectors? Practical geometry turns abstract theorems into physical blueprints.

Locating the Center of a Circle or Arc: Using perpendicular bisectors of two non-parallel chords:

1. Draw two non-parallel chords $AB$ and $PQ$ across the circular region
2. For chord $AB$: center compass at $A$, radius > half of $AB$, swing arcs both sides; repeat from $B$ → intersection points give perpendicular bisector $DE$
3. Repeat for chord $PQ$ → perpendicular bisector $LM$
4. Intersection of $DE$ and $LM$ = center point $O$

Tangent Construction from Point on Circumference:

1. Draw radius $\overline{OP}$ from center to point $P$
2. Construct $90^\circ$ angle at point $P$ (tangent $\perp$ radius)
3. Extend to form tangent line $PQ$

Tangent Construction from External Point:

1. Join center $O$ to external point $P$
2. Construct perpendicular bisector of $OP$ → midpoint $Q$
3. Draw helper circle centered at $Q$ with radius $OQ$
4. Intersection points $A$ and $B$ with original circle = tangency points
5. Draw $PA$ and $PB$ → two equal tangent lines

Important Exam Note: Always write clear Steps of Construction alongside your drawing.

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

• How do we locate the hidden center of an unmarked circle or arc using intersecting perpendicular chord bisectors?
• How do we construct a perpendicular tangent line through a designated point resting directly on a circle's circumference?
• How do we draw a pair of congruent tangent lines from an external point by constructing a midpoint helper circle?
• How do we document clear, sequential steps of construction using formal technical terms for geometry boards?

Frequently Asked Questions (FAQ)