Problem
Figures reflected in a mirror
A 3 by 3 dot board is given. The top-left dot must be one of the four corners. The quadrilateral must fold onto itself along some line. Find every such quadrilateral.
Reasoning · Grade 5-2 Line-Symmetric and Point-Symmetric Figures
Your answer
Watch out Different positions look like different shapes, but a turn or flip makes them one.
How to solve
Strategy Make a Systematic List — Trying quadrilaterals at random and folding each one would mean testing dozens of shapes and still never knowing whether any were missed. So instead of listing the shapes, I list the possible fold lines. There are only a handful of lines that could be a fold line at all, because folding has to send every corner onto another dot of the board — and in particular it has to send the red dot onto a dot. Once the fold line is fixed, the four corners have almost no freedom left, so I can walk through the lines one at a time and be certain nothing escapes.
1STEP 1
Set up names for the dots, and see what the fold test means
Name the dots and fix what the fold test means.
A line of symmetry is nothing more mysterious than a crease that makes the two halves match. Folding tracing paper over a drawing settles any single case in a second — the hard part is being sure you have found them all.
4.G.A.3Create A Physical Representation2STEP 2
Work out where a fold line can possibly go
A fold line has only a few places to go.
Reflecting a point across a line means going the same distance straight across it, so you can check each candidate line just by measuring the red dot's distance to it and stepping the same amount out the other side. If that spot is not a dot, the line is dead — no fiddling with shapes required.
8.G.A.1Make A Systematic List3STEP 3
Cut the work in half using the board's own symmetry
The board's symmetry halves the work.
Since the rules say a turn or a flip does not make a new answer, any symmetry of the board itself lets you skip half the search. Noticing that early saves doing the same case twice.
8.G.A.2Organize Information In More WaysThe board's own symmetry halves the work, because a shape and its mirror twin are the same shape.
Why?
Flipping the board keeps every length and angle, so a shape found on one side has an exact twin on the other.
🧱Rigid motion preserves length and angleSliding, turning, or flipping a shape lays it exactly onto a copy, so every length and angle stays the same (folding is the flip case).
Why?
Grouping the shapes into mirror pairs puts each one in exactly one group, so only one member of each pair needs drawing.
🧱Disjoint cases addSplit the possibilities so no case overlaps another and none is left out, and the case counts simply add.
4STEP 4
Fold line = the long diagonal from the red dot: four shapes
The long diagonal as mirror gives 4.
opposite corner ∈ {centre, bottom-right} × mirror pair ∈ {{top-middle,middle-left}, {top-right,bottom-left}, {middle-right,bottom-middle}}
Fixing the crease first turns an open-ended hunt into a two-by-three table you can read off. The one case that has to be thrown out is easy to spot: three corners in a straight line make a triangle, not a quadrilateral.
4.G.A.1Make A Systematic List5STEP 5
Fold line = halfway between the first two columns: one new shape
A crease between two columns adds 1.
When the crease passes between the dots instead of through them, no corner can sit on it, so the four corners have to split into two mirror pairs. That is what makes rectangles turn up here rather than kites.
8.G.A.1Make A Systematic List6STEP 6
Fold line = the middle column: one new shape
The middle column adds 1.
{top-middle,centre}, {top-middle,bottom-middle} → red dot, top-middle, top-right collinear → not a quadrilateral
This crease is the one that is easy to overlook, because the red dot ends up two whole steps away at the far corner. Checking the crease lines instead of the shapes is exactly what stops that case being missed.
4.G.A.1Make A Systematic List7STEP 7
The two slanted creases: one new shape
A slanted crease adds 1.
A slanted crease is just as good as an upright one; it only looks harder. Reflecting a dot across it still means stepping the same distance straight across, and on this board the slanted crease is the one that produces the trapezoid.
8.G.A.1Make A Systematic List8STEP 8
Collect the list and check every shape by folding
Altogether that is 7.
Listing the creases first is what makes 'I have them all' a fact rather than a hope: every line-symmetric figure must have a crease, every possible crease was checked, so nothing can have escaped.
8.G.A.2Draw A DiagramAnswer
7 quadrilaterals
4 + 1 + 1 + 1 = 7
The count is the right size: the other three corners are chosen from the eight black dots, which is 56 possible corner sets before any are drawn or folded, so a handful of survivors is what you would expect — and the search was run over creases, of which only seven exist and only five needed real testing. Every shape found was folded and checked, and each of the four crease directions contributes shapes of a different kind — upright creases give the rectangles and squares, the slanted crease gives the trapezoid, and the creases through a corner give the kite and the arrowheads. Two of the seven are dented inwards, which is allowed: a dented quadrilateral still has four corners and four straight sides. One caution worth stating plainly: the workbook supplies six dot boards, so a solver following the book will draw six shapes, but a complete check of every possible crease turns up seven, the extra one being the upright arrowhead on the red dot, the centre dot, the top-right dot and the bottom-middle dot, which folds exactly onto itself along the middle column.
Takeaway
Do not hunt for the shapes — hunt for the fold lines first, because there are only a few of them, and each one tells you exactly which dots the corners can be.
- Set up names for the dots, and see what the fold test means
- Work out where a fold line can possibly go
- Cut the work in half using the board's own symmetry
- Fold line = the long diagonal from the red dot: four shapes
- Fold line = halfway between the first two columns: one new shape
- Fold line = the middle column: one new shape
- The two slanted creases: one new shape
- Collect the list and check every shape by folding