Class 6 Mathematics Chapter 8: Playing with Constructions

Ishaan tries to copy a drawing

Ishaan sees a small picture in his notebook. It has a face with a wavy neck, and two eyes made of gentle curves. He copies it freehand. The lines wobble. One eye ends up fatter than the other. His sister looks at it and asks a useful question. What tool would make the two curves match?

Freehand first, tools after

Drawing by hand is not a waste of time. It shows you the shape and tells you what the hard parts are. Ishaan now knows that his problem is the curves, not the straight bits. A rough sketch is a plan, and a plan is worth making before any tool comes out of the box.

What counts as a curve

In this chapter a curve means any shape a pencil can trace on paper. A wave is a curve. A ring is a curve. Surprisingly, a straight line is also counted as one. So saying a figure is made of curves does not rule out its straight edges at all.

Two tools, two jobs

A ruler does straight work and measures lengths. A compass does the curved work. The compass has a sharp tip on one arm and a pencil on the other. Once you fix how far apart the arms sit, that distance is locked until you change it yourself.

Marking points at a fixed distance

Try this on paper. Mark a point and call it P. Now mark a point 4 cm away from P. Then another, in a different direction, also 4 cm from P. Keep going. After a dozen marks a shape starts to appear, and it is the same shape every time anyone does this.

The marks make a circle

The points crowd together into a ring. That ring is a circle. Nothing was drawn round anything, and no lid was traced. The circle appeared purely because one distance was kept the same in every direction.

Centre and radius

P is the centre of that circle. The distance of 4 cm is its radius. Take any point on the ring and measure back to P, and you get 4 cm again. Take a different point on the ring, and you get 4 cm once more.

Setting the compass

Opening the compass by guesswork gives a circle of unknown size. Instead, lay the ruler flat. Put the tip on the zero mark and move the pencil arm until it sits on the 4 cm mark. The compass is now holding a radius of 4 cm.

One sweep, one circle

Press the tip into the paper at P, keep it there, and swing the pencil right round. The whole ring appears in a single movement. The tip must not slide. If it wanders, the distance changes, and the shape closes up badly.

An arc is part of a circle

Often you do not want the whole ring. Sweep for only a short way and you get an arc. An arc still has a centre and a radius behind it. Two arcs made with the same setting will curve by exactly the same amount, which is what makes a pair of matching curves possible.

A wave on a straight line

Here is the wavy neck Ishaan struggled with. Draw a line AB of 8 cm and mark its middle point X. Each bump is to sit on a 4 cm piece of line. Set the compass to 2 cm, place the tip at the middle of a piece, and sweep half a turn.

Why the radius is 2 cm

A half circle has to reach from one end of its 4 cm piece to the other. That span is the full width of the half circle. The centre sits halfway along it, so the radius is 2 cm. Guess this measurement and the bump lands too tall or too flat.

Curves that bulge the other way

Sweep the first half circle above the line and the second one below it, and the two bumps read as a wave. Same setting, same size, opposite sides. The trick with a wave is the matching, and matching is exactly what a fixed compass setting gives you.

Smaller arcs for an eye

An eye shape is two shallow arcs meeting at their ends. Shallow means the compass tip sits further away from the line than the middle point does. The further off you place the tip, the flatter the arc comes out. Choosing that spot well takes a few tries, so use a pencil and expect to rub out.

What the tools really gave

Ishaan draws his picture again, this time with ruler and compass. The neck waves match. The eyes are twins. Nothing about the shape was beyond him before, but only the tools could repeat a distance exactly, over and over, without his hand having to judge it.

Scenario:

A class draws a rangoli pattern of overlapping circles, all the same size. Nita draws hers freehand and the circles come out uneven. Their teacher gives her a compass set to one opening and asks her not to change it. The second attempt is even all the way round. The class works out what the compass guaranteed. Every point on each circle is the same distance from its centre, and keeping the opening fixed keeps that distance the same for every circle. Nita sees that the tool was not making it neater by accident. It was holding one number constant while her hand could not.

Analysis:

The freehand version failed because the radius drifted. A compass holds one distance fixed, and that is exactly what a circle is.

Activity:

Draw a pattern of six equal overlapping circles with a compass. Do not change the opening, and write what stayed the same.

Title:

A garland of circles

Safety:

The compass tip is sharp, so press it into the paper only, and put it away point down.

What You Notice:

Each circle passes exactly through the centres of its neighbours, because every centre is 3 cm from the next and the radius is 3 cm as well. The overlaps make a chain of petal shapes down the line.

Potters shaping a bowl and bakers trimming a roti both rely on the same rule the compass uses. One point stays put while the edge turns around it.

A bicycle wheel is a working circle. Every spoke is a radius, which is why the rim stays the same distance from the hub as you ride.

A drawn circle has no corners at all. That is why a wheel rolls smoothly and a square never will.

What a compass does — Tip still, pencil travels. — Four words that hold the whole idea. If the tip slips, the distance changes and the ring stops being a circle.

Half circle on a line — Half the line, half the circle. — Put the tip at the middle of the piece of line. The radius is half of it, and one sweep gives the bump.

The shapes we think we know

Ask anyone to draw a rectangle and you get one at once. Ask what makes it a rectangle and the room goes quiet. The picture is familiar but the test is not. In this topic we swap the picture for a test, because a test can be applied to a figure you have never seen before.

The two rectangle properties

Take a four sided figure ABCD. It is a rectangle when two things are true. First, the sides facing each other come in equal pairs, so AB matches DC and AD matches BC. Second, every one of its four corners is a right angle. Both must hold. One alone is not enough.

The square properties

A square is tested in a similar way. All four of its sides must be the same length, and all four corners must be right angles. Notice how close this is to the rectangle test. The corner rule is identical, and only the side rule has been tightened.

Squares live inside rectangles

So hand a square to the rectangle test. Are its opposite sides equal? Yes, since all four are equal. Are the corners right angles? Yes. The square passes, which means every square is a rectangle. The reverse fails, because a rectangle 8 cm by 3 cm has unequal neighbours.

Naming the corners

A rectangle has corners, sides and angles, and we label them with letters. The letters are not free to be scattered. A name lists the corners in the order you meet them as you walk round the edge. Start at any corner you like and walk either way round.

Names that work and names that do not

For one square with corners P, Q, R and S in order round it, PQRS is a name. So are QRSP and SRQP, because they are the same walk begun elsewhere or done backwards. PQSR is not a name. Following those letters makes you cut straight across the middle of the figure.

Turning does not change a shape

Cut a square from paper and turn it a little. It now looks like a diamond standing on a point. Has anything been measured differently? The sides are the same pieces of paper they were, and the corners were not bent. So both square tests still pass and it is still a square.

Eyes against tests

This is why a test beats a first impression. Our eyes are trained by the neat upright squares in books, so a tilted one looks like something else. A test asks about lengths and corners only, and neither of those cares which way up the paper is.

Planning a construction

Now to build one. Suppose a square of side 6 cm is wanted. Think before drawing. Four sides of 6 cm, four right angles. That means one side can be drawn first, and then a right angle raised at one of its ends to hold the next side.

Step one, the base

Rule a straight line and mark P on it. Measure 6 cm along and mark Q. That is one side finished, and it fixes the size of everything that follows. Keep the pencil sharp, because a corner marked thickly can be half a millimetre out before you start.

Step two, the right angle

At P, raise a line that stands at a right angle to PQ. This upright line is a guide, not a side. It tells you where the next corner has to live, but you do not yet know how far up it that corner sits.

Step three, carry the length

Set the compass to 6 cm, put its tip at P, and sweep a small arc that crosses the upright guide. Where the arc cuts the guide is exactly 6 cm from P, so that crossing is the corner S. This is the neat part. The compass has carried a length rather than a reading.

Step four, close the figure

Repeat at Q. Raise a right angle there and carry 6 cm up it to find R. Join S to R and the square is closed. Check your work against the tests. Measure SR, which should also be 6 cm, and check the corners at S and R.

Rectangles, and one that cannot exist

A rectangle is built the same way, with the arc set to the second measurement rather than to the first. Try instead to build a four sided figure with four right angles whose opposite sides differ. Every attempt fails. The fourth side arrives too long or too short, and the figure will not close.

Scenario:

A class is asked to construct a square of side 6 centimetres. Arjun draws four lines of 6 and finds the ends do not meet. Their teacher asks what a square needs besides equal sides. The class lists it: four equal sides and four right angles. Arjun had the first and not the second, so his shape leaned over into a rhombus. He draws it again using a set square at each corner and it closes exactly. The class writes the lesson down. Equal sides alone do not fix a shape, because the corners can still move while the lengths stay the same.

Analysis:

Four sides of 6 centimetres describes a family of shapes, not one. The right angles are what pick the square out of that family.

Activity:

Cut four strips of equal length and pin them into a square. Push one corner and write what changes and what does not.

Tile setters check a floor by making sure the corners are square. One bad corner and every tile after it drifts out of line.

A door frame is built as a rectangle on purpose. If the corners drift away from right angles, the door starts to stick.

A rotated square often looks like a diamond to us. Our eyes react to the tilt, but no measurement in the figure has changed at all.

Rectangle and square tests — Pairs and corners, then all four. — A rectangle wants equal pairs and right corners. A square wants the same corners and all four sides equal.

Valid names — Walk the edge, never cut across. — If your finger has to leave the boundary to follow the letters, that order is not a name for the figure.

Two points on a loose leash

Draw a rectangle ABCD with AB of 7 cm and BC of 4 cm. Put a point X on the side AD, wherever you please. Put a point Y on the side BC, again wherever you please. Each point is free to slide along its own side and may even stop right on a corner. The question is what happens to the length of XY.

Guess before you measure

Ask yourself where the two points would be closest, and where they would be furthest apart. A guess costs nothing and makes the measuring interesting. Most people guess the second part correctly and the first part wrongly, which is exactly why the measuring is worth doing.

Keeping track of what you try

Test several positions and write them down in three columns. The distance of X from A, the distance of Y from B, and the length of XY. Without a table the results turn into a jumble, and the pattern hiding inside them never shows itself.

The level position

Slide X down 1 cm from A, and slide Y down 1 cm from B. The join XY now runs level across the rectangle. Measure it. It reads 7 cm, the same as AB. Try both drops at 2 cm and measure again. Still 7 cm.

Why level gives 7 cm

Look at the four sided figure ABYX that you have just made. Its corners at A and B are right angles, and the two drops are equal, so AX matches BY. That makes ABYX a rectangle in its own right, and XY is the side facing AB. Facing sides of a rectangle are equal, so XY reads 7 cm.

Level is as near as they get

Now break the level position. Leave X at 1 cm and slide Y down to 3 cm. The join tilts, and it measures more than 7 cm. Any tilt at all costs length. So the smallest that XY can ever be is 7 cm, which is the length of AB.

Pushing them apart

For the furthest position, drive each point as far from the other as it can go. X slides all the way down to the corner D, and Y climbs all the way up to the corner B. The join DB now stretches from one corner of the rectangle to the opposite corner.

That longest join has a name

A line joining opposite corners of a rectangle is a diagonal. In this rectangle it measures a little over 8 cm, longer than any side. So the two points roam between 7 cm at their nearest and about 8 cm 1 mm at their furthest, and never leave that range.

A different kind of problem

Here is the second task of this topic. Construct a rectangle that can be cut into three identical squares in a row. Notice what has not been given. There is no length anywhere in the question, so the first job is not drawing at all.

Draw the rough figure first

Sketch the finished shape freehand, with the two cutting lines inside it. Label the corners. Now mark the lengths you know must be equal by putting a small tick on each of them. The ticks do the thinking, because they show at a glance which lengths are locked to which.

What the ticks reveal

Every square has four equal sides, and the squares are identical to one another. Follow the ticks round and you find that all the short lines carry the same mark. The long side of the rectangle is made of three of those short lines laid end to end.

Choosing the free measurement

So only one measurement is actually free. Pick the short side, say 4 cm. The long side is then three short sides, which comes to 12 cm. Pick 3 cm instead and the long side would be 9 cm. The shape of the finished rectangle is fixed either way.

Which rectangles refuse to split

Not every rectangle can be cut this way. A rectangle of 5 cm by 8 cm cannot hold two identical squares, because 8 cm is not two lots of 5 cm. A rectangle of 4 cm by 10 cm cannot hold three, since 10 cm is not three lots of 4 cm.

Carrying the length instead of measuring

There is a neat way to build the rectangle without reading the ruler again. Draw the short side and open the compass to fit it exactly, tip on one end and pencil on the other. That length now rides in the compass, and stepping it along the long side three times marks every cut for you.

Scenario:

A class measures the two diagonals of several rectangles cut from card. In every case the two come out equal. Meena says that is a coincidence of their cutting. Their teacher asks the class to test a rhombus as well, where the sides are equal but the corners are not square. Its diagonals differ. So the equality is not true of all four-sided shapes. The class folds one rectangle along a diagonal and finds the two halves match exactly. Meena sees that the right angles are doing the work, because they make the two halves of the rectangle identical.

Analysis:

The rhombus is what rules out coincidence. Equal diagonals belong to the rectangle's right angles, not to four-sided shapes in general.

Activity:

Cut three rectangles of different shapes and measure both diagonals of each. Then try a rhombus and compare.

Carpenters call the corner to corner measurement the diagonal, and they check both of them before fixing a frame. A difference warns them the frame has leaned.

A rectangle split into three identical squares is a favourite shape for a bookmark. It folds into three neat sections of the same size.

The shortest way across a rectangle between its two long sides is always straight across. Any slant adds length without gaining ground.

Nearest and furthest — Level is nearest, corners are furthest. — Equal drops give the shortest join. Opposite corners give the longest, which is a diagonal.

Splitting into squares — Count the short sides along the long one. — Three squares in a row means three short sides lying along the long side of the rectangle.

Two lines across a rectangle

Draw a rectangle PQRS and join P to R. Then join Q to S. Those two lines are the diagonals, and each of them runs from a corner to the corner facing it. Before measuring anything, make a prediction. Are the two diagonals the same length, or is one of them longer?

They come out equal

Measure them. In a rectangle 6 cm by 4 cm both read about 7 cm 2 mm. Try a long thin rectangle instead and measure again. The two readings still match each other. Whatever the shape of the rectangle, its two diagonals are equal in length.

And they cut each other in half

Mark the point where the diagonals cross. Measure from that crossing out to each of the four corners. All four readings are the same, around 3 cm 6 mm in the rectangle above. So each diagonal has been cut into two equal pieces at the crossing.

The centre, found for free

That crossing is the centre of the rectangle. It needed no measuring at all. Two ruled lines found the point that sits the same distance from every corner, which is exactly what you want when a figure has to be placed centrally inside another.

What the square adds

Now repeat the whole experiment on a square. The diagonals are still equal, and they still halve each other. Something extra shows up at the crossing. The angle there is a right angle. In a square the two diagonals meet squarely, and in a rectangle that is not a square they do not.

Do not overclaim

This is worth stating carefully. A rectangle 6 cm by 4 cm has its diagonals crossing at about 113 degrees and about 67 degrees, and neither is a right angle. Saying that the diagonals of any rectangle are perpendicular is a mistake, and it is the commonest one in this topic.

Angles at a corner

A diagonal also does something at the corners it touches. It arrives at a right angle and splits it into two smaller angles. Those two must add to 90 degrees, since together they fill the corner. In a long rectangle one of them is small and the other large.

A construction from those angles

Suppose a rectangle is wanted in which a diagonal splits the corner angles into 60 degrees and 30 degrees. That pair adds to 90 degrees, so it is a legal request. Draw the base, set off a line at 60 degrees from one end, and the diagonal is fixed before any side length has been chosen.

A side and a diagonal

Here is a second construction. Build a rectangle whose side is 5 cm and whose diagonal is 7 cm. Draw the 5 cm base DC first. Raise a line at a right angle to it at C. The corner B lies somewhere on that upright line, but where?

Let the compass find it

B must be 7 cm from D, since that is the diagonal. So set the compass to 7 cm, put the tip at D, and sweep an arc. Every point of that arc is 7 cm from D. Where the arc cuts the upright line is the one point that satisfies both conditions, and that is B.

Finish and check

From B and D, raise the last two right angles to find the fourth corner A. Then test the figure. The sides facing each other should be equal, and all four corners should be right angles. Measure the other diagonal too. It should also read 7 cm.

The house, and a new question

A drawing of a house needs its roof point A. It has to be 5 cm from B and also 5 cm from C, the two top corners of the walls. Hunting for it with a ruler means trying, measuring, and trying again. There is a better way, and it is the arc trick once more.

Two arcs, one crossing

Set the compass to 5 cm. Sweep an arc from B, then another from C without changing the setting. Where the arcs cross is 5 cm from B and 5 cm from C at the same time. No trial, no error. The point simply appears where the two conditions meet.

A whole line of such points

Change the setting to 6 cm and repeat, and a new crossing appears above the first. Every crossing you find lands on one straight line. That line cuts the join from B to C exactly in half and stands at a right angle to it, which is why the arcs always meet along it.

Scenario:

The village of Baradi has two clusters of houses, one near the old banyan tree and one near the school. The panchayat has money for a single hand pump. Both clusters want it close by, and the meeting turns into an argument. Nazia, who is in Class 6, asks for a map and a pair of compasses. She marks the banyan tree as B and the school as C. Then she sets her compass to one span of the map scale and sweeps an arc from each mark, keeping the setting untouched. The arcs cross at a point above the road. She repeats the work with a wider setting and gets a second crossing. Both crossings lie on one straight line, and she draws it.

Analysis:

Nazia turned a fairness argument into a construction. Every point on her line is the same distance from both clusters, so the panchayat can choose among them for good ground.

Activity:

Mark two dots 10 cm apart on paper, find three points equally far from both with different compass settings, and check that all three lie on one straight line.

A carpenter measures both diagonals of a door frame before fixing it. If the two readings differ, the frame has leaned and is no longer a rectangle.

Cricket ground staff mark the pitch by swinging two tapes of equal length from two pegs. The crossing point is their next corner.

A kite maker relies on the square's extra property. Two sticks crossing at a right angle through their middles give a perfectly square outline.

Diagonals of rectangles and squares — Equal and halved always, square corner only for squares. — Equal lengths and halving happen in every rectangle. The right angle at the crossing is the square's own extra.

Finding an equally distant point — Same setting, two centres, one crossing. — Keep the compass setting, sweep from each point in turn, and the crossing is the place you want.