Symmetry, reflection and construction

What this heading covers

Symmetry is the part of geometry most candidates assume they know and most lose a mark on, because the question is almost never "is this symmetrical". It is "how many lines of symmetry does it have", and that is a count people guess.

Construction is asked in words rather than on paper: which instruments a given construction needs, or what the steps produce. You are not asked to draw in the exam. You are asked what drawing it would involve, which is a teacher's question rather than a pupil's.

Line symmetry

A figure has line symmetry (also called reflection or mirror symmetry) if a line can be drawn so that one half is the mirror image of the other. That line is the axis of symmetry.

The counts are asked directly, so hold them:

Shape	Lines of symmetry
Scalene triangle	0
Isosceles triangle	1
Equilateral triangle	3
Rectangle	2
Square	4
Parallelogram	0
Rhombus	2 (its diagonals)
Kite	1
Regular pentagon	5
Regular hexagon	6
Circle	infinitely many

Two of these are where marks go.

A parallelogram has no line of symmetry. It looks as though a diagonal should be one, and a diagonal is not: folding along it does not put one half onto the other. This is the single most frequently missed count in the table.

A rectangle has two, not four. Its lines are the two through the midpoints of opposite sides — not the diagonals. Fold a rectangle along its diagonal and the halves do not match.

The pattern for regular polygons is simple: a regular polygon of n sides has n lines of symmetry. A regular octagon has eight.

Rotational symmetry

A figure has rotational symmetry if it looks the same after being turned through less than a full turn about its centre. The order is the number of positions in a full turn at which it matches itself.

• Square — order 4
• Rectangle — order 2
• Equilateral triangle — order 3
• Parallelogram — order 2
• Regular hexagon — order 6
• Circle — infinite order

The angle of rotation is 360° ÷ order. A square matches every 90°; an equilateral triangle every 120°.

Note the parallelogram again. It has no line symmetry but rotational symmetry of order 2 — the two kinds are independent, and the question that pairs them is testing exactly that. A figure can have one, both or neither.

Every figure returns to itself after a full 360° turn, so order is never zero; a figure with no rotational symmetry is said to have order 1.

Reflection

Reflecting a figure in a line gives a congruent figure — the same size and shape — at the same perpendicular distance on the other side of the line. What changes is orientation: the image is laterally inverted, which is why a reflected letter R reads backwards.

On coordinates, the two reflections asked for are:

• In the x-axis: (x, y) becomes (x, −y).
• In the y-axis: (x, y) becomes (−x, y).

Reflecting in the x-axis changes the sign of y. That is the opposite of what most people guess first, so it is worth saying to yourself the long way: the x-axis is horizontal, a point reflected across it moves up or down, and up-or-down is the y coordinate.

Reflection preserves length and angle. It does not preserve the sense of the figure, which is the difference between it and a rotation.

Construction

The instruments are a straight edge or ruler, a pair of compasses, and a protractor. What the question tests is which of them a construction needs.

A construction is done with compasses and straight edge only. If the question says "construct", the protractor is not assumed. If it says "draw an angle of 47°", that needs a protractor, because 47° is not constructible with compasses alone.

The three constructions expected:

• Perpendicular bisector of a segment — compasses set to more than half the length, arcs from both endpoints above and below, join the two crossings. The line produced is perpendicular to the segment and cuts it in half, which is why it is called both things.
• Bisector of an angle — an arc from the vertex cutting both arms, then equal arcs from those two points, then join the crossing to the vertex.
• Perpendicular to a line from a point — arcs to cut the line at two points equidistant from the foot, then the perpendicular bisector of that cut.

Which angles are constructible with compasses alone follows from being able to bisect: 60° (one arc), and then 30°, 15°; 90° and then 45°, 22.5°; and sums such as 75° or 105°. 40° is not, because 120° cannot be trisected with these tools — a question asking which angle cannot be constructed is asking this.

A question of the kind the paper sets

Which of these has rotational symmetry of order 2 but no line of symmetry?

Work the two properties separately.

• A rectangle has order 2, but it has two lines of symmetry. Out.
• A square has order 4. Out.
• An equilateral triangle has order 3. Out.
• A parallelogram has order 2, and no line of symmetry at all.

The answer is the parallelogram, and it is the answer precisely because the two kinds of symmetry come apart there. Any candidate who treats "symmetrical" as one property cannot reach it.

Mistakes that cost marks here

• Giving a parallelogram a line of symmetry along its diagonal. It has none.
• Giving a rectangle four lines of symmetry. It has two; the diagonals are not axes.
• Treating line symmetry and rotational symmetry as the same thing. A figure can have either without the other.
• Reflecting in the x-axis by changing the sign of x. It changes the sign of y.
• Assuming a protractor is allowed in a construction. "Construct" means compasses and straight edge.
• Saying a figure has rotational symmetry of order 0. The minimum is 1.
• Counting a rhombus's lines of symmetry as 4. It has 2 — the diagonals. The square has 4.

In one line

A parallelogram has no line of symmetry but rotational symmetry of order 2; a rectangle has two axes and not four; reflection in the x-axis flips the sign of y; and "construct" means compasses and straight edge, with 60° and 90° and their bisections the angles that follow.

For the classroom

Line symmetry is one of the few topics where folding settles an argument faster than explaining does. Give children a paper parallelogram and ask them to find the fold that matches the halves. They will try the diagonal, it will not work, and that failure teaches the count better than the table.

For rotational symmetry, a shape cut out and pinned at its centre with a drawing pin can be turned while the class counts the positions where it fits its own outline. Order stops being a number to remember and becomes a number they have just counted.