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Eyes, Arcs and the Perpendicular Bisector
Geometric Constructions · Inductive
Students who arrive at this construction for the first time should handle compass and ruler carefully — the most common error is setting unequal compass widths for the two arc pairs without realising it, which produces a line that does not truly bisect the segment. Watching for wobbling compass settings before students begin will save confused results. Figure it Out Q3, section 14.1 p.140 ("explore whether the two arcs of one intersecting pair must use the same radius....") is a useful self-study item for curious students.
0–8Introduce the 'Eyes' design from Grade 6: draw a segment XY on the board, place compass at X and Y with equal radii and draw crossing arcs to form a lens shape. Ask students to notice that the two arc-centres are equidistant from both X and Y — that observation anchors the idea of equidistance and the perpendicular bisector.
8–20Students draw their own segment XY, then draw equal-radius arcs above and below to find the two intersection points, and join them. They measure distances from each intersection to X and to Y, confirming equal distances, and identify that the joining line bisects XY at right angles.
20–32Figure it Out Q1, section 14.1 p.140 ("explore whether the arcs above and below XY must have the same radius....") — students try unequal radii in pairs and compare results, then frame a brief written justification for why equal radii are necessary.
32–40Figure it Out Q2, section 14.1 p.140 ("explore whether both pairs of arcs must be on opposite sides....") — a quick paired exploration; students sketch both-same-side attempts and record whether the arcs still intersect usefully, then share one finding with the class.
Homework
Explore whether the two arcs of one intersecting pair must use the same radius, and justify your answer. (**Figure it Out Q3, section 14.1 p.140**)
Figure it Out Q1, section 14.1 p.140 — When constructing the perpendicular bisector, explore through construction whether the arcs above and below XY must have the same radius, then justify your answer.
Describe, step by step, how to construct the perpendicular bisector of a segment PQ of length 6 cm using only a ruler and compass. State the compass radius you would use and explain why the two arc-pairs must be drawn on opposite sides of PQ.
Open the compass to any radius greater than half of PQ (i.e., greater than 3 cm). Place the compass at P and draw arcs above and below PQ. Without changing the radius, place the compass at Q and draw arcs above and below PQ so they cross the first arcs. Label the two intersection points A (above) and B (below), and draw the line AB. The arcs must be on opposite sides because if both pairs are on the same side they may not intersect at two distinct points, and even if they do, the line through those points does not cross PQ at its midpoint with a right angle.
Also saved as 10 periods or 14 periods.
- 2Proving Why the Perpendicular Bisector Works
Geometric Constructions - 3Constructing a 90° Angle and Rope Methods
Geometric Constructions - 4Bisecting Angles and Constructing 45°
Geometric Constructions - 5Justifying Angle Bisection and the 8-Petal Design
Geometric Constructions - 6Copying Angles and Building Parallel Lines
Geometric Constructions - 7Constructing Parallel Lines in Many Orientations
Geometric Constructions - 8Trefoil and Pointed Arches — Design Constructions
Geometric Constructions - 9The 60° Angle, Regular Hexagon, and Why It Works
Geometric Constructions - 10Stars, Illusions and the Perpendicular from a Point
Geometric Constructions - 11Mixed Construction Practice — Angles, Arches and Figures
Geometric Constructions - 12Tangrams and 2×1 Tile Puzzles
Tiling - 13Colouring Arguments — When Tiling Is Impossible
Tiling - 14Tiling the Plane — Shapes, Patterns and Nature
Tiling - 15Grid Parity, Removed Squares and Tiling Arguments
Tiling - 16Creating and Analysing Original Tiling Designs
Tiling - 17Four Problems, Two Strands — Constructions and Tiling at Work
Synthesis
- 1. Large Numbers Around Us
- 2. Arithmetic Expressions
- 3. A Peek Beyond the Point
- 4. Expressions Using Letter-Numbers
- 5. Parallel and Intersecting Lines
- 6. Number Play
- 7. A Tale of Three Intersecting Lines
- 8. Working with Fractions
- 9. Geometric Twins
- 10. Operations with Integers
- 11. Finding Common Ground
- 12. Another Peek Beyond the Point
- 13. Connecting the Dots
- 14. Constructions and Tilings
- 15. Finding the Unknown
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