RBSE Class 6 Maths Chapter 8 Playing with Constructions Solutions

Practicing RBSE Class 6 Maths Solutions and Class 6 Maths Chapter 8 Playing with Constructions Solutions Question Answer helps develop logical thinking and accuracy.

Playing with Constructions Class 6 Solutions

Ganita Prakash Class 6 Chapter 8 Solutions Playing with Constructions

Figure it Out (Page 191)

Construct this.

As the length of the central line is not specified, we can take it to be of any length.
Let us take AB to be the central line such that the length of AB is 8 cm. We write this as AB = 8 cm.
Here, the first wave is drawn as a half circle.

Question 1.
What radius should be taken in the compass to get this half circle? What should be the length of AX?
Solution:
Here, AB = 8 cm. Since the ‘wavy wave’ has two equal half circles, we have AX = XB.
∴ X is the mid point of AB.
∴ AX = \(\frac{8}{2}\) = 4 cm, the length of AX is 4 cm.
Let M be the mid point of AX.
∴ AM = MX = \(\frac{4}{2}\) = 2 cm
The centre of the half circle is M.
∴ Radius of half circle = AM = 2 cm
Hence, the radius of the half circle is 2 cm.

Question 2.
Take a central line of a different length and try to draw the wave on it.
Solution:
Step-1. Let us take a central line AB such that AB = 12 cm.

Step-2. Since \(\frac{12}{2}\) = 6 cm, using a ruler, we take a point C on AB such that AC = 6 cm. C is the mid point of AB. As \(\frac{6}{2}\) = 3 cm, using a ruler, take points D on AC and E on CB such that AD = 3 cm and CE = 3 cm. D is the mid point of AC, likewise E is the mid point of CB.

Step-3. With point D as the centre, we draw a half circle of radius 3 cm above the centre line AB. With centre at E, we draw a half circle of radius 3 cm below the central line AB.
We draw vertical lines in the half circles above and below the line AB.

Step-4. The figure above represents the required depiction of the given ‘wavy wave’ with the central line of length 12 cm.

Question 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!
Solution:
We shall draw a ‘wavy wave’ of the form shown in figure-1. Here the waves are smaller than half circle.

Step-1. First we draw a central line AB such that AB = 10 cm.

Step-2. Since \(\frac{10}{2}\) = 2 cm. therefore using a ruler, take point C on AB such that AC = 5 cm. C is the mid point of AB.
Since \(\frac{5}{2}\) = 2.5 cm. using a ruler, take points D on AC and E on CB such that AD = 2.5 cm and CE = 2.5 cm. D is the mid point of AC likewise E is the mid point of CB.

Step-3. Using a protractor, draw perpendiculars at E and D respectively above and below the central line.

Step-4. Using a ruler, we mark points F and G such that DF = 1.5 cm and EG = 1.5 cm.

Step-5. Now, we join AF and GB. With the centre as F, draw an arc from A to C of a radius equal to AF. Similarly with centre as G draw an arc from B to C of a radius equal to GB.

Step-6. We draw vertical lines in fig. 6. also erase the extra lines in fig. 6 as shown in fig. 7.
Fig. 7 represents the required depiction of a ‘wavy wave’ where the waves are smaller than a half circle.

Figure it Out (Page 194)

Question 1.
Draw the rectangle and four squares configuration (shown in Fig.) 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.

Solution:
Step-1. We take a square dot paper and mark a dot on it as A Starting from A we move 10 dots to right and mark the tenth dot as B.

Step-2. Start from B and moving 6 dots above B, we mark the point C. Start from A and moving 6 dots above A, we mark the 6th dot as C, Start from A and move 6 dots above A, we mark the 6th dot as D. By joining AB, BC, CD and DA we get a rectangle ABCD.

Step-3. Now as shown in figure, we take points E, F, G and H on the dot paper.

Step-4. Now we take points I, J, K and L at a distance of 4 dots from E, F, G and H respectively. Now join IE, FJ, GK and LH.

Step-5. On LH and GK, we construct squares above the rectangle.

Step-6. On IE and FJ, we construct squares below the rectangle.

Step-7. The figure given below is the required configuration of one rectangle and four squares on a square dot paper.

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

Solution:
In the fig. A, the number of dots in each side are equal and each angle is 90° therefore it is a square. In the fig. B, lengths of all sides are equal but each angle is not equal to 90°, so it can not be a square. In the fig. C, all sides are of equal length and each angle is of 90°, therefore it is a square.

In the fig. D, the lengths of all 4 sides are not equal therefore it is not a square.

Question 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.
Solution:
We draw three rotated squares and three rotated rectangles on a dot grid such that the comers of squares and rectangles are on dots. Here I, II and III are three squares and IV, V and VI are three rectangles. Using a protractor, we find that all angles are of 90° and by using a ruler we find that in fig. I, II and III, all sides are equal and opposite sides of fig. IV, V and VI are equal.

Construct (Page 197)

Question 1.
Draw a rectangle with sides of length 4 cm and 6 cm. After drawing, check if it satisfies both the rectangle properties.
Solution:
First we draw a rough figure of rectangle ABCD with sides of length 6 cm and 4 cm.

Step-1. Using a ruler, we draw a line AB such that length of AB is 6 cm.

Step-2. Using a protractor, we draw perpendicular lines at A and B.

Step-3. Using a ruler, we mark point D on the perpendicular line at A such that AD = 4 cm. Likewise, we mark point C on perpendicular line at B such that BC = 4 cm.

Step-4. Now we join CD using a ruler and erase the lines above C and D.

Step-5. Using a ruler, we measure the length of CD that is 6 cm. Using a protractor we measure the angles C and D and find that ∠C and ∠D are of 90° each.
We have : (i) AB = CD = 6 cm,
AD = BC = 4 cm
(ii) ∠A = ∠B = ∠C = ∠D = 90°
ABCD is the required rectangle of sides 6 cm and 4 cm.

Question 2.
Draw a rectangle of sides 2 cm and 10 cm. After drawing, check if it satisfies both the rectangle properties.
Solution:
Students should construct the following rectangle as per question 1 :

Here, (i) AB = CD = 10 cm, BC = AD = 2 cm
(ii) ∠A = ∠B = ∠C = ∠D = 90°
ABCD is the required rectangle.

Question 3.
Is it possible to construct a 4-sided figure in which—
• all the angles are equal to 90° but
• opposite sides are not equal?
Solution:
It is not possible to construct a 4-sided figure in which all the angles are equal to 90° but opposite sides are not equal.

Construct (Page 199)

Question 1.
Breaking Rectangles
Construct a rectangle that can be divided into 3 identical squares as shown in the figure.

Solution:
Let us take a rectangle of length 9 cm and breadth 3 cm. This rectangle can be divided into 3 identical squares.

Construction : First we draw a line AB = 9 cm, using a ruler. Now we mark two points P and Q on AB such that AP = 3 cm and PQ = 3 cm. Using protractor, we draw 4 perpendicular lines on A, P, Q and B.

Using ruler we mark points D, S, R, C on perpendiculars such that AD = PS = QR = BC = 3 cm.

Now we join the points C, R, S and D. Thus, we get required rectangle that is divided into three identical squares : APSD, PQRS and QBCR.

Construct (Page 201)

Question 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?

Solution:
The centre of a rectangle (or square) is the intersection point of its diagonals. Using this concept we shall construct the required figure.

Steps of construction :
(1) Using ruler, we draw a line segment AB = 8 cm. Now using protractor, we draw perpendicular lines at points A and B. We mark point P on the perpendicular at A and point Q on the perpendicular at B such that AP = BQ = 4 cm. Now we join points P and Q and also erase the lines above the points P and Q.

(2) Using ruler, we draw diagonals AQ and BP, letting C be their point of intersection. Point C is the centre of the rectangle ABQP and will also serve as the centre for the square to be constructed.

(3) We draw a perpendicular line passing through C, intersecting AB at point R and QP at point S.
Now we erase the diagonals AQ and BP.

(4) Since AP = 4 cm, each side of the square should be 4 cm. Using ruler, we mark points E and D on line AB such that AE = 2 cm and BD = 2 cm and ER + RD = 2 cm + 2 cm = 4 cm.

Likewise, we mark points F and S on PQ such that PF = 2 cm and GQ = 2 cm and FS + SG = 2 + 2 = 4 cm.
(5) Using ruler, we join EF and DG and erase line RS. Thus, EDGF is the required square with centre C within the rectangle ABQP.

Question 2.
Falling Squares (construct)

Make sure that the squares are aligned the way they are shown.
Solution:
Steps to construct falling squares :
(1) First we draw a line seqment AB = 4 cm. Draw a perpendicular line on A and mark point C on it such that AC = 4 cm. At point B draw another perpendicular line and mark point D on it such that BD = 4 cm. Extend line BD to point E such that BD = DE = 4 cm.

(2) We join CD and extend line CD to point F such that DF = 4 cm Using protractor, we draw a perpendicular line on F. Now using a ruler, on perpendicular line on F. we mark two points G and H such that FG – GH = 4 cm Now we join EG.

(3) Now we extend line EG to I such that GI = 4 cm. Using protractor, we draw a perpendicular line on I and mark point J on it such that IJ ‘= 4 cm. Now we join HJ and erase extra lines to get required falling squares.

Now, try this.

Solution:
Students should solve it themselves (following the above procedure).

Question 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.
Solution:
Steps of construction :
(1) First we construct a square ABCD of side 8 cm.
(2) Now we draw lines parallel to the side AB and AD at intervals of 2 cm. dividing the square into 16 smaller squares.
(3) Starting from comer A, we erase the inner sides of four squares to form a large square with a side length of 4 cm, with one comer located at A. Now we draw diagonals in the remaining 12 smaller squares.
(4) Now we draw horizontal lines in the portions of the 12 squares above the diagonals. The resulting figure is the required figure.

Question 4.
Square with a Hole (construct)

Observe that the circular hole is the same as the centre of the square.
Hint: Think where the centre of the circle should be.
Solution:
In the given figure the centre of the square is the same as the centre of the circle. Centre of the square is the point of intersection of its diagonals.

Steps of construction : (1) First we draw a square ABCD with 6 cm side lengths, using ruler and protractor.

(2) By using ruler, we draw diagonals AC and BD by joining points A to C and B to D. Let C be the point of intersection of these diagonals. This point C is the centre of square and circle to be drawn. After finding point C, we erase diagonals AC and BD.

(3) With C as the centre and a radius of 3 cm, we draw a circle inside the square. This is the required square with a hole.

Question 5.
Square with more Holes (construct)

Solution:
In the figure, the centre of a circle is the same as that of the corresponding square.
Steps of construction :
(1) First we draw a square ABCD of 8 Cm side lengths, with the help of ruler and protractor. Now we mark points E, F, G and H on lines AB, BC, CD and DA respectively such that AE = BF = CG = DH = 4 cm. Now we join points H to F and E to G.

(2) Let I be the point of intersection of EG and FH. Now we find the centres of squares AEIH, EBFI, IFCG and HIGD by drawing their respective diagonals. The points where the diagonals of smaller squares intersect are the centres of the circles to be drawn.

(3) Now we erase the diagonals used to locate the centres of the smaller circles and the extra lines as well. Now we draw circles with a radius of 1.5 cm at the points where the diagonals of smaller squares intersect.

Question 6.
Square with Curves (construct)
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!

Solution:
(1) First using ruler and protractor, we draw a square ABCD of 8 cm side lengths. Now we mark points E, F, G and H on lines AB, BC, CD and DA respectively such that AE = BF = CG = DH = 4 cm. Now we join E to G and H to F.

(2) Now we extend the line FH outside the square and mark points I and J on it such that FI = HJ = 4 cm. Likewise, we extend the line EG and mark points K and L on it such that GK = EL = 4 cm. Now we join B to I.

(3) With centres I, J, K and L and a radius equal to BI, we draw four arcs inside the square as shown in the figure. Erasing extra lines, we get the required figure.

Construct (Page 211)

Question 1.
Construct a rectangle in which one of the diagonals divides the opposite angles into 50° and 40°.
Solution:
Rough figure:

Steps of construction :
(1) Using a ruler, we draw a line segment AB = 4 cm.

(2) Using a protractor, we mark dots C and D at angles 50° and 90° (50° + 40°), keeping the central point of protractor at A, as shown in figure.

(3) We draw a perpendicular line to AB at B and let it intersect the extended line AC at E.

(4) We draw a perpendicular line to BE at E and let it intersect the extended line AD at F.
Thus, we get the required rectangle ABEF.

Question 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?
Solution:
Students should do it themselves according to Q. 1.

Question 3.
Construct a rectangle one of whose sides is 4 cm and the diagonal is of length 8 cm.
Solution:
First let us draw a rough figure.

Now following these steps, we construct the required rectangle :
(1) Using ruler, we draw a line AB equal to 4 cm length.

(2) Using protractor, we draw perpendicular lines to AB at A and B.

(3) With the centre at A and a radius equal to 8 cm, we draw an arc to intersect the perpendicular at C. Likewise, with the centre at B and a radius equal to 8 cm, we draw an arc to intersect perpendicular at D. By joining C to D, we get the required rectangle ABCD.

Question 4.
Construct a rectangle one of whose sides is 3 cm and the diagonal is of length 7 cm.
Solution:
Students should do it themselves.

Construct (Page 215)

Question 1.
Construct a bigger house in which all the sides are of length 7 cm.
Solution:
First we draw a rough figure as shown below.

Step-1. Using a ruler, we draw a line AB equal to 7 cm.

Step-2. Using a protractor, we draw perpendicular lines to AB at A and B such that AD = BC = 7 cm.

Step-3. Using a ruler, we take points P and Q on AB such that AP = BQ = 3 cm. Using a protractor, we draw perpendiculars at P and Q of length 2 cm each and mark points R and S as shown.

Step-4. We join R and S. With centres at C and D and a radius of 7 cm, we draw arcs to intersect at point E. Now we join CE and DE. With the centre at E and a radius of 7 cm. we draw an arc from D to C.

It is the required house with all the lines forming the border of house of length 7 cm.

Question 2.
Try to recreate ‘A Person’, ‘Wavy Wave’ and ‘Eyes’ from the section Artwork, using ideas involved in the ‘House’ construction.
Solution:
1. A Person
How will you draw this?

This figure has two components.

You might have figured out a way of drawing the first part. For drawing the second part, see this.

compass in different locations to see which point works for getting the curve. Use your estimate where to keep the tip.

2. Wavy Wave
Construct this.

As the length of the central line is not specified, we can take it to be of any length.
Let us take AB to be the central line such that the length of AB is 8 cm. We write this as AB = 8 cm.
Here, the first wave is drawn as a half circle.

3. Eyes
How do you draw these eyes with a compass?

For a hint, go to the end of the chapter.

Question 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?
Solution:
Yes, a rhombus is a 4-sided figure in which all the sides are equal in length, but it is not a square. It can be constructed by following the given steps :
Step (1) First we draw’ a line segment AB = 6 cm.
Step (2) At point A, we draw a ray AP making an angle 60° with AB. Likewise at point B. we draw a ray BQ making an angle 60° as shown in the figure.
Step (3) Using ruler, we mark points D on AP and C on BQ such that AD = BC = 6 cm.
Step (4) Now we join points C and D. Thus, we get a rhombus with all four sides are of length 6 cm but each angle is not equal to 90° therefore it is not a square.

Playing with Constructions Class 6 Question Answer

Playing with Constructions Class 6 Extra Questions

Multiple Choice Questions—

Question 1.
To construct wavy wave, the first wave is drawn as—
(a) a circle
(b) a line
(c) a half circle
(d) an angle
Answer:
(c) a half circle

Question 2.
A rectangle has all the angles equal to—
(a) 60°
(b) 30°
(c) 75°
(d) 90°
Answer:
(d) 90°

Question 3.
A circle has—
(a) one centres
(b) two centres
(c) three centres
(d) infinite
Answer:
(a) one centres

Question 4.
All points on a circle are equidistant from the centre, this distance is called—
(a) Circumference
(b) Line
(c) Chord
(d) Radius
Answer:
(d) Radius

Question 5.
A figure bounded by four lines is called a square if—
(a) each side is equal
(b) each angle is 90°
(c) each side is equal and each angle is 90°
(d) None of the above
Answer:
(c) each side is equal and each angle is 90°

Fill in the blanks—

1. On rotating the square the measures of its length and ……………………… do not change.
2. The diagonals of a rectangle are of ……………………… length.
3. A ……………………… can be useful in planning how to construct a given figure.
4. A square can be constructed if the length of its ……………………… is given.
Answer:
1. angle
2. equal
3. rough diagram
4. me side

Write True/False for the following statements—

1. All points on a circle are not equidistant from the centre. (True/False)
2. A circle and all its parts can be constructed without a compass. (True/False)
3. Ruler and protractor are required to construct a square. (True/False)
4. The diagonals of a rectangle are of equal length. (True/False)
Answer:
1. False
2. False
3. True
4. True

Very Short Answer Type Questions—

Question 1.
Which instrument is used to draw a perpendicular at any point on a line?
Solution:
Protractor.

Question 2.
A square of side length 5 cm is rotated. Now what is the shape of the figure?
Solution:
Rotated square is still a square as rotating a square does not change its length and angles.

Question 3.
Construct a square with a side of 4.5 cm.
Solution:

Question 4.
Write the properties of a rectangle.
Solution:
(i) The opposite side are equal in length.
(ii) All the angles are 90°.
(iii) Diagonals are equal in length.

Question 5.
Draw a circle of radius 4 cm and show all its parts.
Solution:

Short Answer Type Questions—

Question 1.
Write all the names for the square given below. Write properties of a square.

Solution:
Names for the given square—
(1) ABCD
(2) BCDA
(3) CDAB
(4) DABC
(5) ADCB
(6) BADC
(7) CBAD
(8) DCBA
A square satisfies the following two properties:
(i) All the sides are equal, and
(ii) All the angles are 90°.

Question 2.
How will you draw this figure (A person)?

Solution:
This figure has two components.

For drawing the first part, we draw a circle using compass and draw a line segment at a point on circumference.

For drawing the second part, we fix a radius in the compass and tty placing the tip of the compass in different location to see which point works for getting the curve and find out where to place the tip of the compass.

Essay Type Questions—

Question 1.
Write all the steps while constructing a square of 6 cm.
Solution:
Step 1

First we draw a line segment PQ such that length of PQ is 6 cm using a ruler.
Step 2

We mark a point to draw a perpendicular to PQ through P.
Step 3
Method 1

We mark S on the perpendicular such that PS = 6 cm using a ruler.

Method 2
This can also be done using a compass.

Step 4
We draw a perpendicular to line segment PQ through Q.

Step 5
If we had used the compass, then the next point can easily be marked using it!

Step 6

The side RS is 6 cm and ∠R = ∠S = 90°.

Question 2.
Construct a rectangle one of whose sides is 3 cm and the diagonal is of length 7 cm.
Solution:
Rough figure :

Steps of construction :
(1) First we draw a line segment AB equal to 3 cm.
(2) Using protractor, we draw perpendicular AP at A and perpendicular BQ at B.
(3) With the help of compass, placing the tip of compass at A, taking a radius of 7 cm cut an arc on BQ and obtained point C. Likewise, placing the tip of compass at B, taking a radius of 7 cm cut an arc on AP and obtained point D.
(4) Joining points C and D, we get required rectangle ABCD.