Class 7 Mathematics Chapter 14: Constructions and Tilings

Cutting a line exactly in half

Draw a line segment XY of 8 cm. Now find its midpoint without measuring.

A ruler would do it, but only as well as your eyes can read a scale. A compass can do it exactly.

The construction

Open the compass to 6 cm. Put the point on X and draw an arc above the line and another below it.

Now move the point to Y, keep the same opening, and draw two more arcs. Each pair of arcs crosses at a point.

Joining the crossings

Call the two crossing points A and B. Draw the line AB.

That line cuts XY exactly in half, and it crosses XY at a right angle. It is called the perpendicular bisector of XY.

What an arc really is

To see why it works, remember what an arc means. Every point on an arc is the same distance from the compass point.

So the arcs drawn from X hold all the points 6 cm from X. The arcs from Y hold all the points 6 cm from Y.

So each crossing is equally distant

A crossing lies on both arcs at once. So it is 6 cm from X and also 6 cm from Y.

Both A and B have that property. Each is equally far from the two ends.

Why the line bisects

Now think about the two halves of the picture. The point A is equally far from X and from Y.

So is B. The whole line AB is a mirror line for the segment, which means it must cross XY at its midpoint, 4 cm from each end.

Why the angle is a right angle

The same symmetry gives the angle. Folding along AB lands X exactly on Y.

The two angles at the crossing swap places and must therefore be equal. Two equal angles on a straight line are 90 degrees each.

Equal distance is the real definition

That gives a better way to describe the line. The perpendicular bisector of XY is the set of all points equally far from X and from Y.

Any point on it is equally distant from both ends. Any point equally distant from both ends lies on it.

A check on the drawing

That description also gives a check. Pick any point on your line and measure its distance to X and to Y.

The two measurements should match. If they do not, the line is not the perpendicular bisector.

The opening must be large enough

One step can fail. Open the compass to 3 cm and try the same construction on an 8 cm segment.

The arcs from X reach only 3 cm, and those from Y reach only 3 cm. Together they cover 6 cm of an 8 cm gap, so they never meet.

The rule for the opening

So the opening must be more than half the segment. For 8 cm, anything above 4 cm will do, and 6 cm is a comfortable choice.

Exactly half is not enough either. Arcs of 4 cm would touch at the midpoint only, giving one point rather than two.

Why two crossings are needed

One point is not enough to draw a line. Two points fix a line exactly.

That is why the arcs are drawn both above and below. The upper crossing and the lower crossing give the two points.

Finding a midpoint

The construction gives the midpoint as a free gift. It is where AB crosses XY.

For the 8 cm segment that point is 4 cm from each end, and no measuring was needed to find it.

Where this is used

A perpendicular bisector answers any question about being equally far from two places. A path equally distant from two wells runs along it.

It is also the first step in many other constructions, because it produces a right angle without a protractor.

One habit to keep

Never change the compass opening once the first arc is drawn. The whole argument rests on both distances being the same.

A slipped opening still produces crossings, and the line through them still looks convincing. But it is no longer the perpendicular bisector of anything.

What carries forward

Equal arcs from both ends of a segment cross at two points, and the line through them is the perpendicular bisector. It works because every point on an arc is a fixed distance from its centre, so each crossing is equally far from both ends.

That makes the line a mirror for the segment, which is why it bisects and why the angle is a right angle. The compass opening must be more than half the segment, or the arcs never meet.

Scenario:

A village has two wells, X and Y, 8 metres apart on a plan. A path must run so that every point on it is equally far from both wells. A pupil measures to find the midpoint with a ruler, and draws a line through it by eye. A second pupil uses a compass but opens it to 4 metres on the plan scale. A third opens it to 6 metres but draws arcs only above the line. A fourth draws all four arcs, but widens the compass slightly before moving to Y.

Analysis:

One method is inexact, one opening gives a single point, one drawing has too few points, and one changes the distance the argument needs.

Activity:

Mark two points 8 cm apart, construct the perpendicular bisector, then measure from three different points on it to each end.

Title:

Construct and test

Safety:

Take care with the compass point, and keep it away from your other hand.

What You Notice:

At 3 cm the arcs never meet, and at 4 cm they touch at one point only. At 6 cm they cross twice, and the line through those crossings cuts XY at 4 cm from each end. Every point you test on the line is the same distance from X as from Y.

The construction gives a right angle without a protractor, which is why it begins so many other constructions.

The perpendicular bisector is better described as all the points equally far from two ends than as a list of four steps.

Changing the compass opening partway breaks the argument, because the crossing is then nearer one end than the other.

Open wider than half, or nothing crosses.

Equal arcs, equal distances.

Halving an angle without a protractor

Draw an angle of 76 degrees. Now cut it into two equal parts without measuring anything.

A protractor would give 38 degrees twice, as well as your eyes can read it. A compass does it exactly.

The construction

Put the compass point on the corner. Draw an arc crossing both arms of the angle, and mark the two crossings.

Now put the point on each crossing in turn. Draw two arcs that meet inside the angle, keeping the same opening for both.

Joining to the corner

Call that meeting point P. Draw the line from the corner through P.

That line cuts the angle into two equal halves. It is called the bisector of the angle.

Two triangles appear

To see why, look at what has been drawn. The corner, the first crossing and P form a triangle. The corner, the second crossing and P form another.

Those two triangles share the line from the corner to P.

Three pairs of equal sides

Now compare their sides. The two crossings are the same distance from the corner, because one arc made them both.

The point P is the same distance from each crossing, because the same opening made both arcs. And the line to P belongs to both triangles.

So the triangles are congruent

That is three pairs of equal sides. By the SSS criterion the two triangles are congruent.

Congruent triangles have equal angles in corresponding places. The two angles at the corner are such a pair, so they are equal.

And the angle is halved

Those two equal angles together make the whole 76 degrees. So each is 38 degrees.

No measuring was needed. The construction is exact, and the reason is a congruence.

Bisecting again

The bisector of a bisector halves it again. Bisecting the 38 degrees gives 19 degrees.

So from one angle you can reach a half, a quarter, an eighth, and so on, as far as your pencil allows.

Making the arcs long enough

One practical point. The first arc must actually reach both arms, so draw it generously rather than as a short tick.

The two inner arcs must be long enough to cross as well. If they stop short of each other, there is no meeting point to join, and the opening needs widening.

Copying an angle

Now a different job. You have an angle of 54 degrees in one place and want an identical one somewhere else.

Draw a line where the copy is to sit. This will be one arm of the new angle.

The construction for copying

Put the compass on the original corner and draw an arc across both arms. Without changing the opening, draw the same arc from the new corner.

Now measure the gap between the two crossings on the original, using the compass. Mark that same gap on the new arc.

Joining the mark

Draw a line from the new corner through that mark. The new angle equals the old one.

Again the reason is a congruence. Two triangles have been built with three pairs of equal sides, so SSS makes them congruent and the angles match.

Why copying matters

Copying lets an angle be moved without a protractor and without knowing its size. The 54 degrees never had to be measured.

That is useful whenever a shape must be repeated, which is exactly what the next topic needs.

What you can make from nothing

Some angles can be built with no angle to start from. A perpendicular gives 90 degrees, and bisecting that gives 45 and then 22.5.

An equilateral triangle gives 60 degrees, and bisecting gives 30 and then 15.

What you cannot

But not every angle can be created this way. There is no ruler and compass construction for an angle of 20 degrees from nothing.

What you can always do is COPY an angle you already have, and HALVE any angle at all. Those two are enough for most work.

A check on both constructions

Measure the result with a protractor afterwards. A bisected 76 should give 38 twice, and a copied 54 should give 54.

The protractor is not doing the construction. It is confirming it, which is a different thing.

What carries forward

To bisect an angle, draw one arc across both arms, then equal arcs from the two crossings, and join their meeting point to the corner. To copy one, carry the arc and the gap between its crossings to a new corner.

Both work because the compass builds two triangles with three pairs of equal sides, so SSS makes them congruent and the angles must match. Any angle can be copied or halved, though not every angle can be built from nothing.

Scenario:

A carpenter has a bracket with an angle of 76 degrees and needs it cut into two equal halves. An apprentice measures with a protractor and marks 38 degrees by eye. For a second bracket, the carpenter needs an angle of 54 degrees copied onto a new piece of wood. The apprentice measures it, writes down 54, and sets a protractor on the new piece. He then asks whether a 20 degree brace can be constructed the same way, with no angle to copy from.

Analysis:

Two jobs can be done exactly without any measuring, and one request asks for something the method cannot give.

Activity:

Draw three angles, bisect each one by construction, then measure both halves with a protractor to confirm.

Title:

Bisect, then copy

Safety:

Take care with the compass point. No other hazards.

What You Notice:

Both halves measure 38 degrees, and bisecting again gives 19. The copied angle measures 54 without ever being set with the protractor. Changing the opening between the inner arcs gives two unequal parts, which is exactly what the SSS argument predicts.

Bisection is exact while a protractor reading is not, so the protractor is better used to confirm a construction than to produce one.

Copying an angle never requires knowing its size, which is why it works on an angle nobody has measured.

Changing the opening between the two inner arcs leaves a drawing that still looks finished but no longer halves the angle.

One arc out, two arcs in, one line through.

Carry the arc, then carry the gap.

Floors without gaps

Look at a tiled floor. The tiles cover it completely, with no gaps left over and no tile lying on top of another.

Covering a surface that way is called tiling it. Not every shape can do it, and the reason is surprisingly simple.

The test at a corner

Pick any point where tiles meet. The angles of the tiles crowd around that point and go all the way round it.

All the way round is 360 degrees. So the angles meeting at such a point must total exactly 360, or there is a gap or an overlap.

That is the whole test

A gap means they total less than 360. An overlap means they total more.

So asking whether a shape tiles becomes asking whether copies of its angles can add up to 360 exactly. That is arithmetic, not guesswork.

Squares pass

A square has four corners of 90 degrees each. Four of them give 360, since four nines are 36.

So four squares meet neatly at every point, which is why floors are so often square.

Triangles pass too

An equilateral triangle has three angles of 60 degrees. Six of them give 360, because six sixes are 36.

Lay six equilateral triangles round a point and they close up exactly. The pattern runs on forever.

And so do hexagons

A regular hexagon has angles of 120 degrees. Three of them give 360.

Three hexagons at every point is the honeycomb pattern, and it is one of the reasons honeycomb looks the way it does.

Pentagons fail

A regular pentagon has angles of 108 degrees. Three of them give 324, which leaves a gap of 36.

Four would give 432, which is an overlap. There is no whole number of pentagon corners that totals 360, so regular pentagons cannot tile.

Finding the angle of a regular shape

You need the interior angle before you can test a shape. For a regular shape with a corners, each angle is 180 minus 360 divided by a.

For six sides that is 180 minus 60, which is the 120 used above. For eight sides it is 180 minus 45, which is 135.

Octagons fail as well

Two angles of 135 give 270, and three give 405. Neither is 360.

So a floor of regular octagons alone is impossible, however neat the shape looks.

Only three regular shapes work

Test every regular shape this way and the answer is short. Only the equilateral triangle, the square and the regular hexagon tile on their own.

Everything else leaves a gap or an overlap. That is a strong result from a very small amount of arithmetic.

Mixing shapes changes things

Octagons fail alone, but not in company. Two octagons give 270, and a square of 90 fills the rest, since 270 and 90 make 360.

That octagon and square pattern is common on real floors. The test has not changed, only what is allowed to meet at a point.

How close the pentagon comes

It is worth noticing how narrow the pentagon's failure is. Three of its corners reach 324 degrees, and only 36 are missing.

That is a tenth of a full turn. The shape looks as though it ought to work, and very nearly does, which is exactly why the test is needed rather than the eye.

Any triangle at all tiles

Here is a surprise. Every triangle tiles, not only the equilateral one.

Turn a copy of any triangle upside down and join it to the original. The two make a parallelogram, and parallelograms stack in rows without gaps.

Any quadrilateral too

Even stranger, every quadrilateral tiles, including the awkward ones with a dent in them.

Its four angles total 360 degrees. So put one copy of each corner round a point, and the four angles fill it exactly.

Why the test is the key

Notice what did the work in every case. Not the look of the shape, and not trial and error with cut paper.

It was one question asked at a corner: do these angles total 360? Everything else followed from that.

What carries forward

Tiling means covering a surface with no gaps and no overlaps. At every point where tiles meet, the angles must total exactly 360 degrees, and that single test decides everything.

Among regular shapes only the equilateral triangle, the square and the regular hexagon pass it. Pentagons and octagons fail alone, though octagons work alongside squares. And every triangle and every quadrilateral tiles, because of the way their own angles total up.

Scenario:

A family is tiling a veranda. The shop offers four regular shapes. They are squares, equilateral triangles, regular pentagons and regular octagons. The pentagon is the prettiest. The family would like to use it on its own. The shopkeeper says the octagon can be used too. But something else must be bought alongside it. He does not say what. The family also has a box of leftover tiles. Those were cut as plain uneven quadrilaterals. None of them is regular, and one or two have a dent. The family wonders whether they are any use at all.

Analysis:

One test at a meeting point answers every question the family has.

Activity:

Cut out several copies of one uneven quadrilateral and fit them round a point to see the four angles close up.

Title:

Testing shapes round a point

Safety:

Take care with the scissors. No other hazards.

What You Notice:

Four squares, six triangles and three hexagons each close up exactly. Three pentagons leave a gap of about 36 degrees and a fourth will not go in. Two octagons and a square close up, and so do the four corners of the uneven quadrilateral.

The honeycomb pattern is three regular hexagons at every point, which is the only way regular hexagons can meet.

An octagon floor is really an octagon and square floor, since two octagons leave exactly a square hole.

A regular pentagon fails by only 36 degrees, which is the gap left when three of its corners meet.

Three sixty at a point, or it will not fit.

One eighty minus three sixty over the corners.