A grid of Varignon parallelograms of SOME is given in the figure. To begin with, focus only on the 9 green coloured copies of SOME in the figure. We will first place only these 9 copies in the figure. The corresponding parallelograms are shaded in the figure. Place 9 copies of SOME so as to match the way the green coloured copies of SOME are placed in the figure. If done carefully, you will observe that each 4-gon you placed meets other placed copies exactly at vertices. Now see how 4 copies of SOME are made automatically in the gaps! Carefully place four more copies of SOME in these gaps. Continuing this process with more copies of SOME will give us the desired tiling.
Step-by-step solution
Idea: A quadrilateral can be rebuilt from its midpoint parallelogram by doubling through the midpoints. Since all shaded cells are slid copies of one parallelogram, the copies of SOME on them are slid copies of each other, and the leftover spaces have the shape of SOME turned round.
- Placing a copy: put a copy of SOME on a shaded cell so that the midpoints of its sides S O, O M, M E, E S fall on the corners of the cell, the same way round for every cell. Each copy then pokes out of its cell by four corner triangles.1 mark
- Copies meet at vertices: neighbouring shaded cells (one cell apart) are slid copies of each other along the diagonals SM and OE, so the copy on the next cell is SOME slid along SM (or OE). Vertex M of one copy is then exactly vertex S of the next: they touch at a corner only.1 mark
- The gaps: each gap is surrounded by four placed copies, which supply one side each: SO, OM, ME and ES. Its corners are four vertices of copies, and its angles are 1, 2, 3, 4 again, so the gap is a copy of SOME turned through 180°. Placing these four copies and continuing in every direction tiles the plane.1 mark
Answer to write in the exam
Place each copy with the midpoints of its sides on the corners of a shaded cell (same orientation)
Shaded cells are slid copies along SM and OE ⇒ placed copies are slid copies ⇒ meet only at vertices
Each gap has sides equal to SO, OM, ME, ES (one from each neighbour) and angles 1, 2, 3, 4
∴ Each gap is a copy of SOME turned through 180°; filling gaps and repeating gives the tiling.
Common mistakes that cost marks
- Placing copies on neighbouring cells (side by side) instead of every other cell. The tile-cells alternate with cells that are made of four corner triangles.
- Turning some of the 9 copies. All copies on the shaded cells point the same way; only the gap copies are turned.
- Matching only the corners of SOME to the grid. It is the midpoints of the sides that sit on the grid corners.
How this can come in the exam
The quadrilateral formed by the midpoints of the sides of a quadrilateral tile has area 18 cm². The area of the tile is
- 9 cm²
- 18 cm²
- 36 cm²
- 72 cm²
Show answer
(C) 36 cm²
The midpoint parallelogram has half the area of the quadrilateral (each corner triangle is a quarter of a triangle cut off by a diagonal), so the tile has area 2 × 18 = 36 cm².
Try one yourself
Why does the midpoint parallelogram of a quadrilateral have half its area?
Show answer
Diagonal AC splits ABCD into ∆ABC and ∆ACD. The corner triangles at B and D are each ¼ of these triangles (sides halved), so together they remove ¼ of ABCD; the same for the corners at A and C with diagonal BD. So ½ of ABCD is left.
More questions like this
- The following shape— the first of its kind!— that leads to aperiodic tilings was discovered only recently in 2023 by a team of four mathematicians (Smith, Myers, Kaplan and Goodman-Strauss). Cut out 15 identical copies of the following shape (known as the ‘hat’) and start to tile with it. Do you see any pattern?
- Justify why the plane cannot be tiled with a regular pentagon. (Hint: Read the first 3 sentences of ‘Think and Reflect’ in the section on tiling.) (There are many ways to tile the plane using a suitable irregular pentagon. The most recent method was found in 2015.)
- 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?
- 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.
All Quadrilaterals and parallelograms questions · All maths questions