NCERT Solutions for Class 6 Maths Chapter 8 - Playing with Constructions Exercise 8.4

Exercise 8.4

Question 1. Breaking Rectangles
Construct a rectangle that can be divided into 3 identical squares.

Solution: We shall draw a rectangle of the form shown in Fig. 1.

Step 1. Let us keep the vertical side of the rectangle to 3 cm. Since the rectangle is to be divided into three identical squares, the length of the rectangle must be 3 cm + 3 cm + 3 cm = 9 cm.

Step 2. Using a ruler, draw a line AB equal to 9 cm. (Fig. 2).

Step 3. Using a ruler, find points P and Q on AB such that AP = 3 cm and PQ = 3 cm. Here, QB is also 3 cm. (Fig. 3).

Step 4. Using a protractor, draw perpendicular lines at A, P, Q, and B. (Fig. 4).

Step 5. Using a ruler, mark points A’, P’, Q’, and B’ on perpendiculars at A, P, Q, and B respectively such that AA’ = PP’ = QQ’ BR’ = 3 cm. (Fig. 5).

Step 6. Join A’ and P’, P’ and Q’, and Q’ and B’ using a ruler. Erase the lines above A’, P’, Q’, and B’. (Fig. 6).

Step 7. ABB’A’ is the required rectangle which is divided into 3 identical squares APP’A’, PQQ’P’, and QBB’Q’.

Question 2. Give the lengths of the sides of a rectangle that cannot be divided into: (Page 201)
(i) Two identical squares
(ii) Three identical squares.
Solution:
(i) Let the smaller side of a rectangle be x cm. If the larger side of the rectangle is 2x cm (x cm + x cm), then this rectangle can be divided into two identical squares of side x cm. (Fig. 1)

Let us consider a rectangle of sides 4 cm and 6 cm. Here, 6 is not equal to 8 (4 + 4), so, it cannot be divided into two identical squares as shown in Fig. 2

(ii) Let the smaller side of a rectangle be x cm. If the larger side of the rectangle is 3x cm (x cm + x cm + x cm), then this rectangle can be divided into three identical squares of side x cm. (Fig. 3)

Let us consider a rectangle of sides 3 cm and 8 cm. Here, 8 is not equal to 9 (3 + 3 + 3), so, it cannot be divided into three identical squares as shown in Fig. 4.

Figure it Out:

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?

Hint: Draw a rough figure. What will be the sidelength of the square? What will be the distance between the corners of the square and the outer rectangle?
Solution: The centre of a rectangle (or square) is the point of intersection of its diagonals.

Step 1. Using a ruler, draw a line AB equal to 8 cm. Using a protractor, draw perpendicular lines at A and B. Using a ruler, mark point P on the perpendicular line at A such that AP = 4 cm. Using a ruler, mark point Q on the perpendicular line at B such that BQ = 4 cm. Join P and Q using a ruler. Erase the lines above P and Q. (Fig. 1)

Step 2. Draw diagonals AQ and BP, using a ruler. Let the diagonals intersect at C. This point is the centre of the rectangle ABQP and of the required square. (Fig. 2)

Step 3. Erase diagonals AQ and BP. Using a protractor, draw a perpendicular line on AB and pass through the centre C. Let this perpendicular meet AB at R and PQ at S. (Fig. 3).

Step 4. Since AP = 4 cm, each side of the square must be 4 cm. Using a ruler, mark points A’ and B’ on AB such that A’R = 2 cm and RB’ = 2 cm. Thus, A’B’ = A’R + RB’ = 2 cm + 2 cm = 4 cm.

Similarly, using a ruler, mark points P’ and Q’ on PQ such that P’S = 2 cm and SQ’ = 2 cm. Thus, P’Q’ = P’S + SQ’ = 2 cm + 2 cm = 4 cm. (Fig. 4).

Step 5. Using a ruler, join A’ and P’ and also B’ and Q’. Erase the line RS. (Fig. 5).

Step 6. In Fig. 5, A’B’Q’P’ is the required square with centre C, which is also the centre of the given rectangle.

Question 2. Falling Squares
Construct the ‘Falling Squares” figure shown below:

Make sure that the squares are aligned the way they are shown. Now, try this.

Solution: In the given figure, there are three falling squares and the side of each square is 4 cm.

Step 1. Using a ruler, draw a line AB equal to 4 cm. Using a protractor, draw perpendicular lines at A and B.

Using a ruler, mark point C on a perpendicular line at A such that AC = 4 cm.

Using a ruler, mark points D and E on a perpendicular line at B such that BD = 4 cm and DE = 4 cm. (Fig. 1).

Step 2. Join C and D. Produce CD to F such that DF = 4 cm. Using a protractor, draw a perpendicular line at F. Using a ruler, mark points G and H on a perpendicular line at F such that FG = 4 cm and GH = 4 cm. (Fig. 2).

Step 3. Join E and G. Produce EG to I such that GE = 4 cm. Using a protractor, draw a perpendicular line at I. Using a ruler, mark point J on the perpendicular line at I such that IJ = 4 cm. Join H and J. Erase extra lines in the figure. (Fig. 3).

Step 4. Fig. 3 is the required figure of three “falling squares” each of side 4 cm.

Question 3.

Shadings
Construct the figure given below. Choose the measurement of your choice. Note that the larger 4-sided figure is square and so tire the smaller ones.

Solution:
Step 1. Using a ruler, draw a line AB equal to 8 cm. Because, 8 ÷ 4 = 2, we shall draw smaller squares of side 2 cm. Using a protractor, draw perpendicular lines at A and B. Using a ruler, mark point P on the perpendicular line at A such that AP = 8 cm. Using a ruler, mark point Q on the perpendicular line at B such that BQ = 8 cm. Join P and Q using a ruler. Erase the lines above P and Q (Fig. 1).

Step 2. On the lines AB, BQ, QP, and PA, mark points at distances of 2 cm, using a ruler. Draw horizontal lines and vertical lines to get 16 squares. (Fig. 2)

Step 3. From comer A, erase the inner sides of four squares to get a square of side 4 cm with one comer at A. Draw parallel diagonals of the remaining 12 small squares of side 2 cm each. (Fig. 3)

Step 4. In the 12 small squares, draw horizontal lines in the portion above the diagonals. (Fig. 4)

Step 5. Fig. 4 is the required figure having 12 small squares in a square.

Question 4. Square with a Hole

Observe that the circular hole is the same as the centre of the square.
Construct a “Square with a Hole” as shown in the given figure. The centre of the hole is the same as the centre of the square.
Hint: Think where the centre of the circle should be.
Solution: The centre of a square is the point of intersection of its diagonals. This centre is also the centre of the hole in the figure.

Step 1. Using a ruler, draw a line AB equal to 5 cm, say. Using a protractor, draw perpendicular lines at A and B. Using a ruler, mark point P on the perpendicular line at A such that AP = 5 cm. Using a ruler, mark point Q on the perpendicular line at B such that BQ = 5 cm. Join P and Q using a ruler. Erase the lines above P and Q (Fig. 1).

Step 2. Draw diagonals AQ and BP using a ruler. Let the diagonals intersect at C. This point is the centre of the square ABQP. Erase the diagonals AQ and BP. (Fig. 2).

Step 3. With centre at C and a radius of 1.5 cm, say, draw a circle using a compass. (Fig. 3)

Step 4. Fig. 3 is the required “Square with a Hole”.

Question 5. Square with more Holes
Construct a “Square with Four Holes” as shown in the given figure.

Solution: In the figure, the centre of a circle is the same as that of the corresponding square.

Step 1. Using a ruler, draw a line AB equal to 8 cm, say. Using a protractor, draw perpendicular lines at A and B. Using a ruler, mark point P on the perpendicular line at A such that AP = 8 cm. Using a ruler, mark point Q on the perpendicular line at B such that BQ = 8 cm. Join P and Q using a ruler. Erase the lines above P and Q. (Fig. 1)

Step 2. Using a ruler, find points C, D, E, and F such that AC = 4 cm, BD = 4 cm, QE = 4 cm, and PF = 4 cm. Join C and E and also F and D. (Fig. 2)

Step 3. Let G be the intersection of lines FD and CE. Find the centres of squares ACGF, CBDG, DQEG, and GEPF by joining their respective diagonals. (Fig. 3)

Step 4. Erase the extra lines used for finding the centres of the smaller circles. With centre at centres of small squares, draw four circles of radius 1.3 cm, say. (Fig. 4)

Step 5. Fig. 4 is the required “‘Square with Four Holes”.

Question 6. Square with Curves
This is a square with 8 cm side lengths.

Construct a “Square with Curves ”, taking a square of side 8 cm as shown in the figure.
Hint: Think where the tip of the compass can be placed to get all 4 arcs to bulge uniformly from each of the sides. Try it out!
Solution: In the given figure, the centres of the four arcs are outside the square.

Step 1. Using a ruler, draw a line AB equal to 8 cm. Using a protractor, draw perpendicular lines at A and B. Using a ruler, mark point P on the perpendicular line at A such that AP = 8 cm. Using a ruler, mark point Q on the perpendicular line at B such that BQ = 8 cm. Join P and Q using a ruler. Erase the lines above P and Q. (Fig. 1)

Step 2. Using a ruler, mark points C, D, E, and F such that AC = 4 cm, BD = 4 cm, QE = 4 cm, and PF = 4 cm. Join C and E and also D and F. Extend these lines outside the square. (Fig. 2)

Step 3. Extend DF and take points G and H on it so that DG and FH are equal to 4 cm. Extend CE and take points I and J on it so that Cl and EJ are equal to 4 cm. The distance 4 cm can be taken slightly less than or greater than 4 cm. Join B and G. (Fig. 3)

Step 4. With centres at G, H, I, and J and a radius equal to BG, draw four arcs inside the square as shown in the given figure. Erase the extra lines. (Fig. 4).

Step 5. Fig. 4 is the required “Square with Curves” with the square of side 8 cm.

Benefits of NCERT Solutions for Class 6 Maths Chapter 8 Exercise 8.4 Playing with Constructions

• The solutions provide detailed explanations of geometric constructions, helping students grasp the fundamental concepts clearly.

• Each construction problem is presented with step-by-step instructions, making it easier for students to follow and replicate the processes.

• By practising geometric constructions, students develop practical skills in using a compass and ruler, which are essential tools in geometry.

• The solutions are aligned with the CBSE Class 6 Maths syllabus, ensuring that all necessary topics related to constructions are covered comprehensively.

• Regular practice with these solutions enhances students’ ability to approach and solve geometric problems effectively.

• Engaging with constructions fosters spatial reasoning and visualisation skills, which are important in mathematics and real-life applications.

Class 6 Maths Chapter 8: Exercises Breakdown

Important Study Material Links for Class 6 Maths Chapter 8 - Playing with Constructions

Conclusion

The NCERT Solutions for Class 6 Maths Chapter 8 Exercise 8.4, Playing with Constructions, are an invaluable resource for students seeking to master geometric constructions. With clear, step-by-step guidance and practical applications, these solutions enhance understanding and confidence in using essential tools like a compass and ruler. Aligned with the CBSE syllabus, they provide comprehensive coverage of the necessary concepts, preparing students for more advanced topics in geometry. The availability of FREE PDF downloads ensures easy access for effective study and revision. Overall, these solutions empower students to develop strong foundational skills in geometry, paving the way for future mathematical success.

Chapter-Specific NCERT Solutions for Class 6 Maths

The chapter-wise NCERT Solutions for Class 6 Maths are given below. Go through these chapter-wise solutions to be thoroughly familiar with the concepts.

Related Important Links for Maths Class 6

Along with this, students can also download additional study materials provided by Vedantu for Maths Class 6.