NCERT Solutions for Class 9 Maths Chapter 12 — Exercise Set 12.4#

In any tiling, the angles of the tiles that meet around a point must add up to exactly 360° — no gaps and no overlaps.

1. Justify why the plane cannot be tiled with a regular pentagon. #

Solution

Each angle of a regular pentagon is:

$$\dfrac{(5 - 2) \times 180°}{5} = 108°$$

Look at a corner of a tile. Around that point the angles must total 360°.

  • If only pentagon corners meet there: $3 \times 108° = 324°$ leaves a gap of 36°, which no pentagon angle (108°) can fill; and $4 \times 108° = 432°$ is too much. Since $\dfrac{360}{108}$ is not a whole number, this is impossible.
  • If the point lies on the side of another tile, that side contributes a straight angle of 180°, leaving 180° for pentagon corners: $108° = 108°$ leaves 72°, and $2 \times 108° = 216°$ is too much. Impossible again.

∴ Regular pentagons cannot meet around a point without gaps or overlaps, so they cannot tile the plane.

2. 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? #

Method 1 : rotating about midpoints
  1. Draw DART (with the dent at $D$) and copy it.
  2. For each side, rotate a copy through 180° about the midpoint of that side. The rotated copy fits exactly along that side, because a half-turn about the midpoint swaps the two ends of the side.
  3. Repeat this at every side of every new copy.

Around each vertex, the four angles of DART (one from each of four copies) meet, and they add up to 360° — even with the reflex angle at $D$ — so the copies fit together with no gaps.

Method 2 : the Varignon grid
  1. Draw the Varignon parallelogram of DART (join the midpoints of its sides) and tile the plane with copies of this parallelogram, as a grid.
  2. Place a copy of DART on every other parallelogram of the grid (like the black squares of a chessboard), matching the way DART sits around its own Varignon parallelogram (Exercise Set 12.3, Q5).
  3. The gaps left over are also copies of DART — fill them in.
Which do you prefer?

Either answer is fine with a reason. Method 1 is easier to carry out with paper cut-outs (just keep turning copies about midpoints). Method 2 is easier to draw neatly, because the grid of parallelograms fixes where every copy goes.

Quick links