Inscribing Squares in a Circle
Construct two perpendicular diameters → their endpoints are the vertices of an inscribed square.
Introduction
Perpendicular diameters → inscribed square
Past Knowledge
Perpendicular bisector (13.1.2). Inscribed hexagons (13.1.4). Central angles.
Today's Goal
Inscribe a square in a circle using perpendicular diameters.
Future Success
Regular octagons, 12-gons (bisecting 90° arcs), coordinate geometry.
Key Concepts
Core Idea
A square has 4 equal sides and 4 right angles → needs 4 equally spaced vertices on the circle → 4 arcs of 90° each → use two perpendicular diameters.
Side Length
The side of an inscribed square = radius × √2 (from the 45-45-90 triangle formed by two radii and a side).
Step-by-Step Construction
Inscribed Square in Circle O
- 1Draw a diameter .
- 2Construct the perpendicular bisector of (which passes through center O). This creates a second diameter .
- 3Connect the four endpoints: A–C–B–D → inscribed square.
Why It Works
The perpendicular diameters create four 90° central angles. Equal central angles intercept equal arcs, so all four chords (sides) are congruent.
Each inscribed angle subtends a semicircle (180° arc) → each corner angle = 90°.
Four equal sides + four 90° angles = square. ✓
∎ Perpendicular diameters → four 90° arcs → equal chords + right angles → square.
Common Pitfalls
Connecting Wrong Vertices
Connect adjacent endpoints (A→C→B→D), not opposite ones. Connecting A→B and C→D just gives you the diameters again!
Real-Life Applications
Clock Faces
12, 3, 6, and 9 form an inscribed square. Clock designers use perpendicular diameters to place the hour markers.
Square Pegs & Round Holes
The largest square that fits in a circle has side s = r√2. This determines the maximum square peg for a round hole.
Practice Quiz
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