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Playing with Constructions — Class 6 Mathematics

Ganita Prakash · Chapter 8

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Chapter summary

This chapter introduces geometric construction using a ruler and a compass. Starting from freehand artwork, students discover that all points at a fixed distance from a point form a circle, where is the centre and the fixed distance is the radius. The compass is used to draw circles and arcs of given radius. The chapter then revisits squares and rectangles and their defining properties: a rectangle has opposite sides equal and all angles (R1 and R2), while a square additionally has all four sides equal (S1 and S2). Rotating a square or rectangle does not change these properties. Students construct squares and rectangles of given side lengths by drawing perpendiculars and transferring lengths with a compass, and learn to name figures by going around their corners in order. They construct rectangles from a side and a diagonal, and rectangles whose diagonals divide the corner angles into given parts, discovering that equal division happens only for a square. The diagonals of a rectangle are shown to be equal, and the chapter closes with locating points equidistant from two given points (used to build figures like a house) and constructing four-sided figures with all sides equal.

Artwork with Ruler and CompassCircle, Centre and RadiusSquares and Rectangles and their PropertiesConstructing Squares and RectanglesDiagonals of Rectangles and SquaresPoints Equidistant from Two Points

Key ideas & terms

Circle: The set of all points at a fixed distance from a centre point.
Radius: The fixed distance between the centre of a circle and any point on the circle.
Compass: A geometry instrument used to draw circles and arcs of a chosen radius.
Rectangle properties: A rectangle has opposite sides equal (R1) and all angles (R2).
Square properties: A square has all sides equal (S1) and all angles (S2).
Diagonal: A line segment joining two opposite corners of a rectangle or square.

Extra questions & answers

What shape is formed by all points at a fixed distance from a point ?

A circle, with as its centre and the fixed distance as its radius.

What is the radius of a circle?

The distance between the centre of the circle and any point on the circle.

What two properties define a rectangle?

Its opposite sides are equal in length (R1) and all four of its angles are (R2).

What extra property makes a square different from a rectangle?

A square has all four sides equal, in addition to all angles being .

Does rotating a square change whether it is a square?

No; rotation does not change side lengths or angles, so a rotated square is still a square.

How can a rectangle be validly named?

By listing its corners in the order you travel around it, starting from any corner, e.g. ABCD or BCDA.

Which instrument helps draw a perpendicular and transfer equal lengths in constructions?

A compass, together with a ruler, is used to draw arcs, transfer lengths and locate points.

When does a diagonal divide a rectangle\u2019s corner angles into two equal parts?

Only when the rectangle is a square (all four sides equal).

How are the two diagonals of a rectangle related in length?

The two diagonals of a rectangle are equal in length.

Is there a four-sided figure with all sides equal that is not a square?

Yes; a rhombus has all sides equal but its angles need not be , so it is not a square.

Think

Imagine marking all the points of 4 cm distance from the point P. How would they look?

All points that are exactly from lie on a circle with centre and radius . If you join them you get a circle.

Figure it Out

1. What radius should be taken in the compass to get this half circle? What should be the length of AX?

The half-circle covers half of . So which is the diameter of the half-circle. The radius to be taken is half the diameter, i.e. .

2. Take a central line of a different length and try to draw the wave on it.

Take any length, for example . Mark its midpoint so that . With radius draw a semicircle above and with the same radius draw a semicircle below . You get a similar wavy wave of total length .

3. Try to recreate the figure where the waves are smaller than a half circle (as appearing in the neck of the figure, ‘A Person’). The challenge here is to get both the waves to be identical. This may be tricky!

Draw a short segment . Choose a radius smaller than half of , for example if take . Place the compass tip above the segment on its perpendicular bisector and draw a gentle arc joining to the middle point, and repeat with the same radius and same height for the second arc. Using the same radius and same centre-height for both arcs makes the two small waves identical.

Which of the following is not a name for this square?

Which of the following is not a name for this square? 1. PQSR

is not a valid name. It jumps diagonally from to and does not go round the square in order, so it is the incorrect name.

Which of the following is not a name for this square? 2. SPQR

goes round the square in order , so it is a valid name.

Which of the following is not a name for this square? 3. RSPQ

goes round the square in order , so it is a valid name.

Which of the following is not a name for this square? 4. QRSP

goes round the square in order , so it is a valid name. Only is invalid.

Figure it Out

1. Draw the rectangle and four squares configuration (shown in Fig. 8.3) on a dot paper. What did you do to recreate this figure so that the four squares are placed symmetrically around the rectangle? Discuss with your classmates.

On dot paper first draw the central rectangle, then count dots to keep equal gaps on all four sides. Placing four congruent squares with one on each side of the rectangle and at the same distance from the rectangle corners makes the whole figure symmetric vertically and horizontally.

2. Identify if there are any squares in this collection. Use measurements if needed.

In the collection of four figures labelled , only figure has all sides equal and all angles . So is a square, the others are not squares ( and are rhombuses and is a rectangle).

3. Draw at least 3 rotated squares and rectangles on a dot grid. Draw them such that their corners are on the dots. Verify if the squares and rectangles that you have drawn satisfy their respective properties.

Any square drawn with corners on dots that has equal side lengths and corners is a correct square, and any rectangle drawn with opposite sides equal and corners is a correct rectangle. Rotation does not change lengths or angles, so the rotated figures still satisfy for squares and for rectangles when measured with ruler and protractor.

Think

Is it possible to reason out if the sides are equal or not, and if the angles are right or not without using any measuring instruments in the above figure? Can we do this by only looking at the position of corners in the dot grid?

Yes. On a dot grid the horizontal and vertical distances can be counted in dot units. If the steps from one corner to the next are the same in dot counting, the sides are equal. Right angles can be seen when one side goes dots right and dots up and the next side goes dots right and dots down, which is a turn. So symmetry and dot positions let us reason without instruments.

Construct

1. Draw a rectangle with sides of length 4 cm and 6 cm. After drawing, check if it satisfies both the rectangle properties.

Draw , erect perpendiculars at and , mark and and join . On measuring, , and , so and are satisfied.

2. Draw a rectangle of sides 2 cm and 10 cm. After drawing, check if it satisfies both the rectangle properties.

Draw base , erect lines at its ends, mark on both, join the top side. Measuring gives opposite sides and and all four angles , so it is a rectangle.

3. Is it possible to construct a 4-sided figure in which— all the angles are equal to 90º but opposite sides are not equal?

No. If all four angles are the figure is forced to have opposite sides equal. A quadrilateral with four right angles is always a rectangle, so opposite sides cannot be unequal.

Construct

Construct a rectangle that can be divided into 3 identical squares as shown in the figure.

Draw a segment of any length, say . Erect a perpendicular at and mark and such that using a compass. Then complete the rectangle so that and . The rectangle of size by is divided into three squares.

Give the lengths of the sides of a rectangle that cannot be divided into two identical squares: two identical squares

Any rectangle where length is not exactly twice the breadth cannot be divided into two identical squares. For example or . Here , so it cannot be split into two equal squares.

Give the lengths of the sides of a rectangle that cannot be divided into three identical squares: three identical squares

Any rectangle where length is not exactly three times the breadth cannot be divided into three identical squares. For example or . Here , so it cannot be split into three equal squares.

Construct

1. A Square within a Rectangle Construct a rectangle of sides 8 cm and 4 cm. How will you construct a square inside, as shown in the figure, such that the centre of the square is the same as the centre of the rectangle?

For a rectangle the centre is the intersection of its diagonals. The inner square must be to fit the breadth. Leave margin on left and right and on top and bottom, so the square is centred. Draw the rectangle, find its centre, then draw the square around the same centre.

2. Falling Squares Now, try this. Square of side 5 cm Square of side 7 cm Square of side 3 cm

Draw three separate squares with sides , and using the standard construction: draw base, erect perpendiculars, mark side length with compass, join the fourth side. Each satisfies and .

3. Shadings Construct this. Choose measurements of your choice. Note that the larger 4-sided figure is a square and so are the smaller ones. Make sure that the squares are aligned the way they are shown. Each is a Square of side 4 cm

Choose the outer square, for example . Inside it place three small squares aligned in a row or as shown, all sides parallel to the outer square. The construction uses perpendiculars and compass transfer of lengths.

4. Square with a Hole Observe that the circular hole is the same as the centre of the square. Hint: Think where the centre of the circle should be.

Construct a square, for example . Its centre is the intersection of its diagonals. With that centre as the compass tip, draw a circle of any suitable radius, say , so the hole is centred.

5. Square with more Holes

Draw the same square and mark several points symmetric about the centre. With each point as centre draw equal circles, for example radius , to get multiple holes arranged symmetrically.

6. Square with Curves This is a square with 8 cm sidelengths. Hint: Think where the tip of the compass can be placed to get all the 4 arcs to bulge uniformly from each of the sides. Try it out!

Construct an square. For each side, place the compass tip at the midpoint of the opposite side or outward on the perpendicular bisector at a fixed distance, e.g. choose radius and draw an inward bulging arc between the two vertices of the side. Repeat with the same radius and same outward distance for all four sides to get four identical uniform bulges.

Construct

1. Construct a rectangle in which one of the diagonals divides the opposite angles into 60° and 30°.

Draw base of any length. At draw a ray making with for diagonal , and at erect a perpendicular. Their intersection is . Then erect perpendiculars at and to get . Then and , making and the figure a rectangle.

2. Construct a rectangle where one of its sides is 5 cm and the length of a diagonal is 7 cm.

Draw base . Erect a perpendicular at . With centre and radius draw an arc intersecting at . Then complete the rectangle by erecting perpendiculars at and to meet at . Then and diagonal .

Construct

1. Construct a rectangle in which one of the diagonals divides the opposite angles into 50° and 40°.

Draw base , at draw diagonal making with , at erect to get on the diagonal, then complete with perpendiculars at and to get . The diagonal splits into and .

2. Construct a rectangle in which one of the diagonals divides the opposite angles into 45° and 45°. What do you observe about the sides?

The same construction with on each side of the diagonal gives a rectangle where all sides are equal, i.e. it becomes a square. So length equals breadth.

3. Construct a rectangle one of whose sides is 4 cm and the diagonal is of length 8 cm.

Draw , erect perpendicular at , with centre radius cut the perpendicular at , then complete the rectangle to . Check that side and diagonal are correct.

4. Construct a rectangle one of whose sides is 3 cm and the diagonal is of length 7 cm.

Draw , erect perpendicular at , with centre radius cut the perpendicular at , then complete the rectangle. The rectangle satisfies side and diagonal .

Construct

1. Construct a bigger house in which all the sides are of length 7 cm.

Construct base and square on it, then locate apex as intersection of arcs of radius from and , join and and draw the roof arc with centre radius joining to . All border lines are .

2. Try to recreate ‘A Person’, ‘Wavy Wave’, and ‘Eyes’ from the section ‘Artwork’, using ideas involved in the ‘House’ construction.

Use the same compass-arc method: for Wavy Wave use midpoints as centres and radius half the segment to get half-circles; for Person use a circle for the head and a square with a top arc whose centre is found by intersecting arcs of equal radius from the top corners; for Eyes place two lens shapes by intersecting arcs from two centres and above and below the eye line with the same radius.

3. Is there a 4-sided figure in which all the sides are equal in length but is not a square? If such a figure exists, can you construct it?

Yes, a rhombus. All four sides equal but angles not . Construct it by drawing a segment , then with centres and and same radius draw arcs intersecting at and on opposite sides of to get a diamond with but angles not , so it is not a square.