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

In NCERT Solutions Class 6 Maths Chapter 8 - Playing with Constructions, Exercise 8.4 teaches students to draw perpendiculars from a point to a line, whether that point is on or off the line. The ability to construct shapes is fundamental for students to develop an understanding of geometric properties, and is a foundation for properties of geometry in future classes. 

NCERT Solutions Class 6 Maths Chapter 8 Playing with Constructions Exercise 8.4 provides detailed step by step explanations and diagrams to help students correctly solve every question with confidence. With these solutions learners can practice and would clear their doubts and improve their exam result.

1.0Download NCERT Solutions of Class 6 Maths Chapter 8 Playing with Constructions Exercise 8.4: Free PDF

Download NCERT Solutions of Class 6 Maths Chapter 8 Playing with Constructions Exercise 8.4, and get clear and detailed step wise answers in a downloadable PDF. And to make the best use of it to help learners understand constructions and improve their geometry skills with ALLEN facilitators.

Key Concepts of Exercise 8.4

• Constructing a perpendicular from a point on a line 
• Drawing a perpendicular from a point off a line 
• Using compass and ruler correctly for geometric constructions 
• Understanding perpendicular and its importance in geometry 
• Applying geometric tools to help solve a problem

2.0NCERT Exercise Solutions Class 6 Maths Chapter 8 Playing with Constructions : All Exercises 

3.0NCERT Class 6 Chapter 8 Playing with Constructions Exercise 8.4 : Detailed Solutions

8.4 An Exploration In Rectangles

(Construct)

1. Breaking Rectangles Construct a rectangle that can be divided into 3 identical squares as shown in the figure.
   
   Sol. We shall draw a rectangle of the form shown in figure below. 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 $3\mathrm{cm}+3\mathrm{cm}+3\mathrm{cm}$ $=9\mathrm{cm}$. Step 2. Using a ruler, draw a line AB equal to 9 cm .
   
   Step 3. Using a ruler, find points P and Q on AB such that $\mathrm{AP}=3\mathrm{cm}$ and $\mathrm{PQ}=3\mathrm{cm}$. Here, QB is also 3 cm .
   
   Step 4. Using a protractor, draw perpendicular lines at $\mathrm{A},\mathrm{P},\mathrm{Q}$ and B .
   
   Step 5. Using a ruler, mark points ${\mathrm{A}}^{\prime },{\mathrm{P}}^{\prime },{\mathrm{Q}}^{\prime }$, and ${\mathrm{B}}^{\prime }$ on perpendiculars at $\mathrm{A},\mathrm{P},\mathrm{Q}$ and B respectively such that ${\mathrm{AA}}^{\prime }={\mathrm{PP}}^{\prime }={\mathrm{QQ}}^{\prime }={\mathrm{BB}}^{\prime }=3\mathrm{cm}$.
   
   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'.
   
   Step 7. ${\mathrm{ABB}}^{\prime }{\mathrm{A}}^{\prime }$ is the required rectangle which is divided into 3 identical squares ${\mathrm{APP}}^{\prime }{\mathrm{A}}^{\prime }$, PQQ'P', and QBB'Q'.

(Construct)

2. 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 side length of the square? What will be the distance between the corners of the square and the outer rectangle? Sol: 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 $\mathrm{AP}=4$ cm . Using a ruler, mark point Q on the perpendicular line at B such that $\mathrm{BQ}=4\mathrm{cm}$. Join P and Q using a ruler. Erase the lines above P and Q.
   
   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.
   
   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 $S$.
   
   Step 4: Since $AP=4\mathrm{cm}$, each side of the square must be 4 cm . Using a ruler, mark points $A^{\prime }$ and ${\mathrm{B}}^{\prime }$ on AB such that ${\mathrm{A}}^{\prime }\mathrm{R}=2\mathrm{cm}$ and ${\mathrm{RB}}^{\prime }=2\mathrm{cm}$. Thus, ${\mathrm{A}}^{\prime }{\mathrm{B}}^{\prime }={\mathrm{A}}^{\prime }\mathrm{R}+{\mathrm{RB}}^{\prime }=2\mathrm{cm}+2\mathrm{cm}=4\mathrm{cm}$. Similarly, using a ruler, mark points ${\mathrm{P}}^{\prime }$ and ${\mathrm{Q}}^{\prime }$ on PQ such that ${\mathrm{P}}^{\prime }\mathrm{S}=2\mathrm{cm}$ and ${\mathrm{SQ}}^{\prime }=2\mathrm{cm}$. Thus, ${\mathrm{P}}^{\prime }{\mathrm{Q}}^{\prime }={\mathrm{P}}^{\prime }\mathrm{S}+{\mathrm{SQ}}^{\prime }=2\mathrm{cm}+2\mathrm{cm}=4\mathrm{cm}$.
   
   Step 5: Using a ruler, join A' and P' and also B' and Q'. Erase the line RS.
   
   Step 6: In $A^{\prime }{B}^{\prime }{Q}^{\prime }{P}^{\prime }$ is the required square with centre $C$, which is also the centre of the given rectangle.
3. Falling Squares
   
   Make sure that the squares are aligned the way they are shown. Now, try this
   
   Sol. 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 $\mathrm{AC}=4\mathrm{cm}$. Using a ruler, mark points D and E on a perpendicular line at B such that $\mathrm{BD}=4\mathrm{cm}$ and DE $=4\mathrm{cm}$.
   
   Step 2. Join C and D . Produce CD to F such that $\mathrm{DF}=4\mathrm{cm}$. 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 $=4\mathrm{cm}$ and $\mathrm{GH}=4\mathrm{cm}$.
   
   Step 3. Join E and G. Produce EG to I such that GI = 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 $=4\mathrm{cm}$. Join H and J . Erase extra lines in the figure.
   
   Step 4. The required figure of three "falling squares" each of side 4 cm .
4. Shading Construct this. Choose measurements of your choice. Note that the larger 4-sided figure is a square and so are the smaller ones.
   
   Sol. Step 1. Using a ruler, draw a line AB equal to 8 cm . As, $8÷4=2$, 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 $\mathrm{AP}=8\mathrm{cm}$. Using a ruler, mark point Q on the perpendicular line at B such that $\mathrm{BQ}=8\mathrm{cm}$. Join P and Q using a ruler. Erase the lines above P and Q .
   
   Step 2. On the lines $\mathrm{AB},\mathrm{BQ},\mathrm{QP}$, and PA , mark points at distances of 2 cm , using a ruler. Draw horizontal lines and vertical lines to get 16 squares.
   
   Step 3. From corner A, erase the inner sides of four squares to get a square of side 4 cm with one corner at A. Draw parallel diagonals of the remaining 12 small squares of side 2 cm each.
   
   Step 4. In the 12 small squares, draw horizontal lines in the portion above the diagonals.
   
   Step 5. The required figure having 12 small squares in a square.
5. 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. Sol. 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 center of the square. 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 $\mathrm{AP}=5\mathrm{cm}$. Using a ruler, mark point Q on the perpendicular line at B such that $\mathrm{BQ}=5\mathrm{cm}$. Join P and Q using a ruler. Erase the lines above P and Q .
   
   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 .
   
   Step 3. With centre at C and a radius of 1.5 cm , say, draw a circle using a compass.
   
   Step 4. The required "Square with a Hole".
6. Square with more Holes Construct a "Square with Four Holes" as shown in the given figure.
   
   Sol. 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 $\mathrm{AP}=8\mathrm{cm}$. Using a ruler, mark point Q on the perpendicular line at B such that $\mathrm{BQ}=8\mathrm{cm}$. Join P and Q using a ruler. Erase the lines above P and Q .
   
   Step 2: Using a ruler, find points $\mathrm{C},\mathrm{D},\mathrm{E}$, and F such that $\mathrm{AC}=4\mathrm{cm},\mathrm{BD}=4\mathrm{cm},\mathrm{QE}=4\mathrm{cm}$, and $\mathrm{PF}=4\mathrm{cm}$. Join C and E and also F and D .
   
   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.
   
   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.
   
   Step 5: The required "'Square with Four Holes".
7. 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! Sol. 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 $\mathrm{AP}=8$ cm . Using a ruler, mark point Q on the perpendicular line at B such that $\mathrm{BQ}=8\mathrm{cm}$. Join P and Q using a ruler. Erase the lines above P and Q.
   
   Step 2: Using a ruler, mark points $\mathrm{C},\mathrm{D},\mathrm{E}$, and F such that $\mathrm{AC}=4\mathrm{cm},\mathrm{BD}=4\mathrm{cm},\mathrm{QE}=4\mathrm{cm}$, and $PF=4\mathrm{cm}$. Join $C$ and $E$ and also $D$ and $F$. Extend these lines outside the square.
   
   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 .
   
   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.
   
   Step 5: The required "Square with Curves" with the square of side 8 cm .

4.0Key Features and Benefits Class 6 Maths Chapter 8 Playing with Constructions : Exercise 8.4

• Comprehensive NCERT Solutions for ease of understanding
• Visual aids included for a visual learner
• Ideally suited for self-study, homework assistance, exam study
• Builds a solid foundation for future study of the subject in geometry
• Facilitates student clarity and confidence, and assists students in completing NCERT solutions class 6 maths chapter 8 playing with constructions Exercise 8.1 and future problems