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?)
Step-by-step solution
Idea: Two copies of any triangle make a parallelogram, and any parallelogram tiles the plane like a slanted grid. Use the test “diagonals bisect each other ⇒ parallelogram”.
- Let M be the midpoint of BC. Turn a copy of ∆ABC through 180° about M: B goes to C, C goes to B, and A goes to a point A′ with M the midpoint of AA′. The copy is ∆A′CB, congruent to ∆ABC.1 mark
- In quadrilateral ABA′C the diagonals AA′ and BC bisect each other at M, so ABA′C is a parallelogram (diagonal test).1 mark
- Copies of this parallelogram tile the plane: two sets of equally spaced parallel lines (parallel to AB and to AC) make a grid of such parallelograms. Draw the diagonal BC in every cell: each cell becomes two copies of ∆ABC, so the triangles tile the plane.1 mark
Answer to write in the exam
M = midpoint of BC; turn ∆ABC through 180° about M ⇒ A → A′, B ↔ C; ∆A′CB ≅ ∆ABC
Diagonals AA′ and BC of ABA′C bisect each other at M ⇒ ABA′C is a parallelogram
Parallelograms tile the plane (grid of lines ‖ AB and ‖ AC)
∴ Each parallelogram = two copies of ∆ABC ⇒ copies of ∆ABC tile the plane.
Common mistakes that cost marks
- Joining two copies along a side by sliding instead of turning. Slid copies do not make a parallelogram.
- Not giving the reason the two copies form a parallelogram (the diagonals bisect each other, or opposite sides are equal).
- Thinking only special triangles (equilateral, right) tile. Every triangle does.
How this can come in the exam
Two copies of a triangle with sides 5 cm, 6 cm and 7 cm are joined along the 7 cm side to form a parallelogram. Its perimeter is
- 18 cm
- 22 cm
- 26 cm
- 36 cm
Show answer
(B) 22 cm
The joined side becomes a diagonal; the parallelogram’s sides are 5, 6, 5, 6 cm, so the perimeter is 22 cm.
Try one yourself
In the triangle tiling, how many triangles meet at each corner point, and why is the total angle 360°?
Show answer
Six triangles meet, bringing each of the angles A, B, C twice: 2(A + B + C) = 2 × 180° = 360°.
More questions like this
- 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.
- You saw how to use the Midpoint Theorem to divide a given triangle into 4 congruent triangles. Can we divide a triangle into 3 congruent triangles? This exercise shows us how to do that and more, provided we are allowed to cut and reassemble.
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