Draw a non-convex 4-gon DART. Show how we can tile the plane with copies of DART. Both methods that we discussed earlier will work. Which do you prefer?
Step-by-step solution
Idea: Nothing in either method needed the quadrilateral to be convex: half-turns about edge midpoints always make neighbours share whole edges, and the four angles (one of them reflex) still add to 360° round every corner.
- Draw a non-convex DART, e.g. D(1.6, 2.1), A(0, 4), R(5, 2.2), T(0.4, 0): D lies inside triangle ART, so the angle at D is reflex. Its angles still add to 360°.½ mark
- Method 1: turn a copy through 180° about the midpoint of each side; the copy fits along that whole side. Repeat for every side of every new copy. At each corner the four angles of DART meet once (the reflex angle fills more than half the turn, the other three fill the rest), so there are no gaps or overlaps (see the picture).1 mark
- Method 2: draw the midpoint parallelogram of DART and the grid of its copies; place DART on every other cell (same way round) so that its side midpoints sit on the cell corners; the gaps are DART turned through 180°.1 mark
- Preference (model answer): Method 1, because it needs only one rule (half-turn about an edge midpoint) and no extra grid. Method 2 is useful for seeing why the pattern repeats.½ mark
Check: Angles of the sample DART: about 30.1° at A, 45.4° at R, 34.7° at T and 249.8° (reflex) at D; total 360° ✓.
Answer to write in the exam
DART: D(1.6, 2.1), A(0, 4), R(5, 2.2), T(0.4, 0); D inside ∆ART ⇒ reflex ∠D; angle sum 360°
Method 1: half-turn copies about midpoints of sides ⇒ whole edges shared; each vertex gets all four angles = 360°
Method 2: grid of midpoint parallelograms; copies on alternate cells; gaps = half-turned copies
∴ Both methods tile the plane; I prefer Method 1 (one simple rule, no grid needed).
Common mistakes that cost marks
- Measuring the outside angle at the dent (less than 180°). The angle used in the tiling is the reflex one.
- Believing a dented shape cannot tile. The angle sum is still 360°, which is all the corners need.
- Giving a preference with no reason. Say why one method is easier or clearer.
How this can come in the exam
A non-convex quadrilateral tile has angles 30°, 40° and 50° at three corners. Find the fourth angle and explain why four copies can still meet round a point.
Show answer
Fourth angle = 360° − 120° = 240° (reflex) (1 mark). One copy of each angle at the point gives 30 + 40 + 50 + 240 = 360°, a full turn, so the copies fit round the point (1 mark).Try one yourself
Can an arrowhead (dart) shape with angles 45°, 45°, 60° and 210° tile the plane?
Show answer
Yes. Its angles add to 360°, and copies turned through 180° about the midpoints of the sides fit together with one copy of each angle at every corner.
More questions like this
- Using a fact about parallelograms, show how to tile the plane using any given triangle. (Hint: Can you use the parallelogram tiling in the introduction?)
- Mark the midpoint of the line drawn on the paper (see the figure), given that the horizontal lines are equally spaced. Justify your answer.
- You know that the sum of angles of a quadrilateral is 360°, even for a non-convex quadrilateral. (Recall the proof.) Now consider a self-intersecting quadrilateral ABCD, where AB and CD intersect at point E. Show that ∠A + ∠B + ∠C + ∠D < 360°. Can you construct ABCD such that ∠A + ∠B + ∠C + ∠D = 2°?
- In a parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP = BQ (see the figure). Show that APCQ is a parallelogram.
- A right-triangle shaped cutout of a paper is folded such that point A touches point B (see the figure). Show that the crease line can be used to find the midpoint of not only AB but also that of AC.
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