Reasoning · Grade 5-2 Using Symmetry

Problem

Cut a point-symmetric figure into congruent halves

Two 3 by 2 rectangles, one at the upper left and one at the lower right, overlap in a 2 by 1 block to make a 10-square figure. All the inside grid lines are dotted. One cut along those dotted lines splits it in two. Find every cut giving two congruent pieces.
Your answer
How to solve
Strategy Make a Systematic List — The half-turn symmetry of the figure is the key that unlocks the whole question. If a cut goes through the centre of symmetry and is itself unchanged by the half turn, then the half turn swaps the two pieces, which proves at once that they are congruent. That reduces the job to a much smaller subproblem: find the paths that run from the centre out to the edge, because each such half-path, together with its half-turn image, is a complete cut. Those half-paths are few enough to list one by one with nothing missed.
1STEP 1

Set up names for the grid points

Give the grid points names.

2STEP 2

Find the centre of symmetry

Find the figure's centre of symmetry.

O = (2, 1.5)
3STEP 3

Why a cut through O gives congruent pieces

A cut through it makes the pieces congruent.

half turn about O: piece 1 ⇔ piece 2
4STEP 4

Reduce the problem to half a cut

So only the half-path needs finding.

number of cuts = number of half-paths from O upward to the outline
5STEP 5

List the half-paths from O

There are 5 half-paths upward.

1 + 1 + 3 = 5
6STEP 6

Draw the five cuts

Draw the five cuts.

cut 1: (2,0) - (2,3) cut 2: (0,1) - (2,1) - (2,2) - (4,2)
7STEP 7

Check the pieces and check that nothing else works

Each piece holds 5 squares.

10 ÷ 2 = 5 squares in each piece
Answer
5 cuts
10 ÷ 2 = 5
Each of the five cuts leaves 5 unit squares on each side, which is the only possible split of 10 squares into two congruent pieces, so the areas are right. Each cut passes through the centre O, and each is unchanged by a half turn about O, so the half turn carries one piece exactly onto the other - that is congruence with no measuring needed. The answer is a small whole number, as it must be, since a path leaving O has only a handful of ways to reach the outline before it runs out of dotted lines; and the book prints six blank copies, one more than needed, so the count being just under the number of frames is not surprising. Checking every way of splitting the 10 squares into two connected groups of 5 turns up these same five and nothing else.
Takeaway

Find the pin the figure spins on, draw only half the cut out to the edge, and let the half turn draw the other half for you - there are just 5 ways to do it.

  • Set up names for the grid points
  • Find the centre of symmetry
  • Why a cut through O gives congruent pieces
  • Reduce the problem to half a cut
  • List the half-paths from O
  • Draw the five cuts
  • Check the pieces and check that nothing else works