Every corner has a twin
Half of a figure is drawn on a dot grid, beside a dashed mirror line. The task is to complete the figure so that the mirror line is a line of symmetry: folded along the line, the new half must land exactly on the half that is drawn.
For that, every corner of the drawn half needs a twin on the other side of the line. The twin is on the same row as the corner and the same distance from the line, but on the opposite side.
The left half is drawn, and it meets the dashed mirror line at two points.
Count, cross, count
Take one corner at a time. Count the gaps between dots from the corner to the mirror line, along its row. The top-left corner is 3 gaps to the left of the line. Cross the line and count the same number of gaps on the other side: its twin is 3 gaps to the right of the line, on the same row.
Do the same for every corner. A corner 1 gap from the line has its twin 1 gap beyond it, and a point that sits on the mirror line is its own twin. Then join the twins in the same order as the corners they copy.
The corner 3 gaps to the left of the line has its twin 3 gaps to the right, on the same row.
The completed figure: each corner on the left has a twin on the right, the same number of gaps from the line.
A mirror line across the grid
The mirror line can also run across the grid. Then count the gaps up and down instead. A corner 2 gaps above the line has its twin 2 gaps below it, in the same column.
The two top corners are 2 gaps above the line, and their twins are 2 gaps below it. The two corners on the line are their own twins.
Flip it, do not slide it
Two slips are common. Counting from the edge of the grid, or from the middle of the drawn half, instead of from the mirror line, puts the twin in the wrong place. And a mirror moves a corner only across the line, never along it: the twin of a corner is always on the same row when the line runs up and down.
The new half faces the other way from the drawn half. A copy slid across the line would face the same way, and it would not fold onto the drawn half.
Worked example: One Fold, One Hole
Question A square sheet of paper is folded in half so that the left edge lands on the right edge. A hole is punched through both layers, 2 cm from the fold and 3 cm from the top edge. When the sheet is unfolded, how many holes are there, and how far apart are they?
1.Two layers, one punch: 2 holes.
Folded left onto right: two layers, one punch. 2.Unfolded, the two holes are mirror images in the fold line, each 2 cm from it and 3 cm from the top.
Unfolded: two holes, mirror images in the fold. 3.Distance between them = 2 + 2 = 4 cm.
Each is 2 cm from the fold: 2 + 2 = 4 cm apart.
Answer: 2 holes; 4 cm apart
Common mistakes
- Placing the second hole 2 cm from the first instead of 2 cm from the fold on the other side.
- Measuring the holes from the top edge of the folded sheet as if it had changed: the fold is vertical, so heights are unchanged.
Worked example: Fold Along a Diagonal, Cut a Corner
Question A square sheet ABCD is folded along the diagonal AC, so that B lands on D. A small triangle is cut off at the folded corner where B and D lie together. When the sheet is unfolded, how many corners have been cut off, and how many sides does the shape now have?
1.Folding along AC puts B exactly on D; the cut goes through both layers.
Folded along AC, corner B lies exactly on corner D. 2.Unfolded, the corners B and D are both cut off; A and C are untouched.
One cut through both layers removes both corners. 3.Each cut replaces a corner with a new short edge: the 4-sided square gains 2 sides.
Unfolded: B and D are gone, A and C untouched. 4.The shape now has 4 + 2 = 6 sides.
Each cut adds an edge: 4 + 2 = 6 sides.
Answer: 2 corners; 6 sides
Common mistakes
- Cutting only one corner in the mind's eye: the cut goes through two layers.
- Thinking the corner A or C is cut: they lie on the fold line, at its ends, not at the folded corner.