Constructions and Tilings · Lesson 12 of 12
Chapter Summary and Practice
“Connect the chapter’s exact constructions, design methods, and tiling arguments through a complete recap and mixed practice.”
• Recall the essential constructions and their justifications. • Choose a construction method from the given information. • Connect angle relationships with hexagons and tilings. • Distinguish a valid tiling from an impossibility proof. • Apply the chapter’s ideas in mixed problems.
Chapter Recap Table
This chapter used equal distances to control exact drawings and careful counting to control coverings. The table revisits every teaching lesson. Use it to identify the right method, then return to a full lesson if you need the complete construction steps.
| Topic studied | Central idea | Method or relationship | Reminder |
|---|---|---|---|
| Eyes and perpendicular bisector | Equal distances locate the bisector | SSS followed by SAS; midpoint and 90° crossing | Both midpoint and perpendicular conditions are needed |
| 90° at a given point | Make that point an auxiliary midpoint | Choose OX = OY, then find one more equidistant point | The method works in any orientation |
| Rope constructions | Taut equal lengths act like compass distances | Rope midpoint gives AX = AY | Use equal usable lengths, allowing for fastening |
| Angle bisection | Divide an angle equally from its vertex | OA = OB, AC = BC, common OC give SSS | Select the internal ray |
| Angle copying | Reproduce an isosceles triangle | Same arc radius and transferred chord give SSS | Transfer the straight chord, not the curved arc |
| Parallel lines | Copy a corresponding angle | Equal corresponding angles establish parallelism | Place the copy in the correct position |
| Arches | Plan supports before drawing curves | Matching lengths, angles, centres, and radii | The outline and the support framework have different roles |
| Regular hexagons | Six equilateral triangles fit around a point | 6 × 60° = 360°; interior angles are 120° | Regular means equal sides and equal angles |
| Related constructions | Reuse established methods | 60° → 30° → 15°; alternate hexagon vertices form stars | An external-point perpendicular uses equidistant line points |
| Tangrams | Rearrange seven fixed pieces | Five triangles, one square, one parallelogram | Use every piece once without gaps or overlap |
| Rectangular domino tilings | Pair rows or columns | An intact m × n rectangle is tileable iff one dimension is even | Do not extend the sufficiency rule to every region |
| Impossible tilings | Every domino covers opposite colours | Unequal checkerboard counts rule out all placements | Equal counts alone do not prove general possibility |
| Plane tilings | Repeat a compatible arrangement | Squares, equilateral triangles, and regular hexagons fit | A finite patch must extend consistently |
Construction Method Checklist
Begin a construction by stating what is given and what must be produced. Then name the equality or angle relationship that controls it. This avoids confusing a copied angle with a bisected angle, or an ordinary perpendicular with a perpendicular bisector.
For a perpendicular bisector, locate two distinct points equidistant from the segment endpoints. For a perpendicular at a known point on a line, first choose an auxiliary segment whose midpoint is that point. For a perpendicular from an outside point, obtain equal-distance endpoints on the line using a circle centred at that point.
For angle bisection, build two matching triangles sharing a side from the vertex. For angle copying, reproduce the three sides of a triangle at another vertex. For a parallel line, place the copied angle in the corresponding position along a transversal. For an arch or decorative figure, construct supports first and apply those methods to matching parts.
Key Relationships
Several small numerical relationships support many of the chapter’s constructions. They become useful when you can say what each quantity represents. Do not use an angle total without identifying the meeting point or use a tile count without identifying the covered region.
| Relationship | Meaning |
|---|---|
| Equal adjacent angles on a straight line are 90° each | Their sum is 180° |
| Eight equal sectors around a point are 45° | 360° ÷ 8 = 45° |
| Halving 60° gives 30°, then halving again gives 15° | Use successive exact bisections |
| Regular-hexagon interior angle is 120° | Two 60° triangle angles meet at each vertex |
| An m × n rectangle has mn squares | m rows of n cells each |
| A valid domino tiling uses mn/2 tiles for an intact rectangle | Each tile covers exactly two cells |
| A domino covers one square of each checkerboard colour | Edge neighbours have opposite colours |
Problem
You need a line through P outside l that meets l at 90°. Which method applies?
- 1.A circle centred at P meets l at X and Y, so PX = PY.
- 2.Construct the perpendicular bisector of XY.
- 3.Because P is equidistant from X and Y, P lies on that bisector.
- 4.The resulting line passes through P and is perpendicular to l. An arbitrary bisector of a different segment might not pass through P.
Problem
A regular hexagon is divided into its six central triangles. Find the central angle and an outer interior angle.
- 1.The central pieces are congruent equilateral triangles, so each centre angle is 60°.
- 2.Six centre angles total 6 × 60° = 360°, filling the turn.
- 3.At an outer vertex, two triangle angles meet, giving 60° + 60° = 120°.
- 4.These equal sides and equal outer angles establish regularity.
Problem
Compare an intact 5 × 6 grid with a fourteen-cell region containing eight white and six black cells.
- 1.The intact grid has an even column count, so horizontal domino pairs cover every row. It uses 30 ÷ 2 = 15 dominoes.
- 2.The other region has even area, so its numerical tile count would be seven.
- 3.However, seven dominoes would cover seven white and seven black cells, not eight and six.
- 4.An explicit strategy establishes the first region’s possibility; unequal colour counts prove the second region impossible.
Common Mistakes
The most useful corrections concern conditions that are easy to overlook. Ask whether the midpoint, equal radii, corresponding angles, and actual tile shape match the method you are applying. A correct result needs the right reason.
A perpendicular does not necessarily bisect a segment. Equal-looking arcs need not have equal radii. Copying the curved arc length is not copying the chord. An even number of cells does not guarantee a general domino tiling, and balanced colours alone do not prove one. A 360° angle fit at one vertex does not by itself establish every proposed plane tiling.
For construction practice, keep supporting lines faint but visible so the reasoning can be checked. For tiling practice, label tile placements or count checkerboard colours. Explain a possibility claim with a valid arrangement and an impossibility claim with an obstruction that applies to every placement.
Check Your Understanding
Use these questions to check the conditions behind each method, as well as the result. For a diagram-based question, follow the labels and given information rather than judging by appearance.
Quiz
Which property is required in addition to a 90° crossing for a perpendicular bisector?
Which construction transfers a chord length to reproduce an angle at another vertex?
What exact angle results from bisecting 30°?
Which intact rectangle cannot be tiled with dominoes?
What proves an eight-white, six-black region impossible to tile by dominoes?
How many regular hexagon corners meet around a vertex in the standard plane tiling?
What is the complete set of tangram pieces?
Practice Problems
- Construct the perpendicular bisector of a 9 cm segment. State a suitable radius and explain why it works.
- Construct a perpendicular at a point on a sloping line, and explain why the chosen point lies on it.
- Bisect a given angle, then copy the original angle at a new point. Explain how the two tasks differ.
- Construct 45°, 30°, and 15°, identifying the starting angle for each method.
- Construct a parallel to a line through an outside point and label corresponding angles.
- Plan a trefoil arch using equal side supports of 6 cm as semicircle diameters. Find the side radii.
- Construct a regular hexagon of side 4 cm and explain its 60° and 120° angles.
- Explain why the six-pointed star’s small point triangles are equilateral.
- If one small tangram triangle has area 4 square units, find the complete set’s area.
- Decide whether 11 × 6 and 7 × 7 rectangles can be tiled, giving a strategy or proof.
- Explain why removing the top-middle square from a 5 × 3 board prevents a domino tiling under the top-left-white colouring.
- Describe the four-L-tile covering of the source’s twelve-cell region.
- Compare square, triangle, and hexagon plane tilings using angles at a meeting point.
- Find a repeating pattern around you and distinguish a geometric observation from a claim that would need further evidence.
Use a radius greater than 4.5 cm, for example 5.5 cm. Two equal-radius crossings are equidistant points, and their joining line bisects the segment at 90°.
Key Takeaways
• Exact constructions depend on stated equalities, not only on appearance. • Perpendicular-bisector reasoning applies to midpoint and external-point constructions. • SSS explains angle bisection and copying, while copied corresponding angles establish parallels. • Equilateral triangles connect 60°, 120°, hexagons, and several designs. • Tiling requires complete coverage with no gaps or overlapping interiors. • Use a constructive arrangement for possibility and a universal obstruction for impossibility. • The whole chapter connects equal distances, exact angles, repeating units, and careful reasoning.
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Tiling the Entire Plane
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