Students can refer to BSE Odisha Class 6 Math Solution and Ganita Prakash Chapter 8 Playing with Constructions Class 6 Question Answer to understand textbook questions step by step.
Class 6 Maths Chapter 8 Playing with Constructions Solutions
Ganita Prakash Class 6 Chapter 8 Solutions
Class 6 Maths Ganita Prakash Chapter 8 Solutions Playing with Constructions
Question 1.
Draw the rectangle and four squares configuration 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:
Draw a rectangle using four dots and draw four squares to make sure all . the squares are equal in size and placed around the rectangle in symmetry.
Question 2.
Take a central line of a different length and try to draw the wave on it.
Solution:
Step 1: Draw a central line AB = 10 cm.
Step 2: Since half of AB is 5 cm, mark a point X on AB such that AX = XB = 5 cm.
Step 3: Mark a point C on AX such that AC = CX =2.5 cm. Mark another point D on XB such that-.. . XD = DB = 2.5 cm.
Step 4: With C as centre and radius equal to AC, draw a semicircle above the line AB, Again, with D as centre and radius equal to BD, draw a semicircle below the line AB.
Step 5: The resultant figure is the required wavy wave with central line of length 10 cm.
InText Questions
Question 1.
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:
Step 1: Draw a line segment AB = 7 cm.
Step 2: At A and B draw perpendiculars with the help of a protractor.
Step 3: fake two points C and D on the two perpendiculars such that AD = 4 cm and BC = 3 cm.
Step 4: Join DC.
Since the opposite sides AD and BC are not equal, it is found that neither ∠D nor ∠C is 90°.
Hence, we conclude that it is not possible to draw a four-sided figure with all angles equal to 90°, when opposite sides are not equal.
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:
Step 1: Draw a horizontal line segment AB (say 5 cm). This will be one side of the rectangle.
Step 2: At point A, use a protractor to measure and draw AX at 45° angle.
Step 3: At point B, measure and draw a 90° angle. Draw a line segment BC extending from B at this angle meeting AX at C.
Step 4: At points A and C, draw a 90° angle which meets at the point D.
Thus, ABCD is the required rectangle.
We observe that all sides are equal. Hence, ABCD is a square.
Playing with Constructions Class 6 Extra Questions
Playing with Constructions Class 6 Very Short Question Answer
Question 1.
Construct a circle with the radius 4.5 cm.
Solution:
Steps of construction of a circle with radius 4.5 cm are as follows:
Step 1: Using a ruler, open the compass to a radius of 4.5 cm.
Step 2: Mark a point O on the paper. Place the pointed end of the compass on point O.
Step 3: Rotate the compass around point 0 to draw a circle ensuring that the pencil remains in contact with the paper at all times.
Question 2.
Construct a square with the side 4 cm.
Solution:
Steps of construction of a square with side 4 cm are as follows:
Step 1: Draw a line segment AB = 4 cm using a ruler.
Step 2: At points A and B, construct two perpendiculars to AB using a protractor.
Step 3: From points A and B, mark points D and C on the perpendiculars respectively such that AD = BC = 4 cm.
Step 4: Join CD to complete the square ABCD.
Question 3.
Construct a rectangle with the side lengths 4 cm and 7 cm.
Solution:
Steps of construction of a rectangle with side lengths 4 cm and 7 cm are as follows:
Step 1: Draw a line segment AB = 7 cm using a ruler.
Step 2: At points A and B, draw perpendiculars to AB using a protractor.
Step 3: From points A and B, mark points D and C on the perpendiculars respectively such that AD = BC = 4 cm.
Step 4: Join CD to complete the rectangle ABCD.
Playing with Constructions Class 6 Short Question Answer
Question 1.
Construct a circle with the following radius:
(i) 4 cm
(ii) 5 cm
(iii) 6 cm
Solution:
(i) Step by step construction of a circle with radius 4 cm:
Step 1: Using a ruler, open the compass to a radius of 4 cm.
Step 2: Mark a point 0 on the paper. Place the pointed end of the compass on point 0.
Step 3: Rotate the compass around point 0 to draw a circle ensuring that the pencil remains in contact with the paper at all times.
(ii) Follow the same steps for radius 5 cm as in (i).
(iii) Follow the same steps for radius 6 cm as in (i).
Question 2.
Recreate the following falling squares:
Solution:
In the figure, we have three identical squares of side length 3 cm.
Step 1: Construct a square ABCD with each side measuring 3 cm.
Step 2: Produce AD to a new point G and CD to a new point E such that DC = DE = 3 cm.
Step 3: Construct square DEFG.
Step 4: Produce EE to a new point J and GF to a A new point H such that FJ = FH = 3 cm.
Step 5: Construct square FHIJ.
Question 3.
Recreate the following falling squares:
Solution:
In the figure, we have three squares of side lengths 5 cm, 4 cm and 3 cm respectively.
Step 1: Construct a square ABCD with each side measuring 5 cm.
Step 2: Produce AD to a new point G and CD to a new point E such that DG = DE = 4 cm.
Step 3: Construct square DEFG.
Step 4: Produce EF to a new point J and GF to a new point H such that FJ = FH = 3 cm.
Step 5: Construct square FHIJ.
Question 4.
Recreate the following combinations of square and hole:
Centre of the hole is the same as the centre of the square containing it.
Solution:
Side length of square is not given. You can take any side length of your choice, say 5 cm.
Step 1: Construct a square ABCD of side 5 cm.
Step 2: Draw diagonals AC and BD. Let O be their point of intersection. Point O is the centre of square ABCD.
Step 3: Using a compass, draw a circle with centre O and radius less than half of the side length of the square ABCD.
The figure so obtained is the required square with a hole.
Question 5.
Construct a rectangle which can be divided into 2 identical squares.
Solution:
A rectangle that can be divided into two identical squares must have one side equal to twice of the other.
Let’s take 3 cm and 6 cm as the side lengths of the rectangle.
Steps of construction:
Step 1: Draw a line segment AB = 6 cm using a ruler, (this will be the longer side of the rectangle).
Step 2: Divide AB into two equal parts. Mark the midpoint M, so that TM = MB = 3 cm.
Step 3: At points A, M and B, draw perpendiculars to AB (using a protractor) long enough to mark the height of the rectangle i.e. 3 cm.
Step 4: Mark points D, N and C on the perpendiculars respectively such that
AD = MN = BC = 3 cm.
Step 5: Join CD to complete the rectangle ABCD.
Here, the rectangle ABCD is divided into two squares (AMND and MBCN) each with the side length of 3 cm.
Question 6.
Recreate the shadings as shown in the figure:
Choose measurements of your choice.
Note: The larger 4 sided- figure is a square and so are the smaller ones.
Solution:
Let the side length of the smaller squares be 2 cm.
Then, the side length of the larger square = 4 × 2 = 8 cm.
Steps of construction are as follows:
Step 1: Construct a square PQRS of side length 8 cm.
Step 2: Using a ruler, mark points at distances 2 cm on sides PQ, QR, RS and SP of square PQRS.
Step 3: Draw horizontal and vertical lines through the marked points to get smaller squares with side equal to 2 cm as shown in the figure.
Step 4: Draw diagonals of the smaller squares.
Step 5: In small squares with side equal to 2 cm, draw vertical lines in the portions below the diagonals as shown in the figure.
Question 7.
Construct a rectangle in which one of its sides is 5 cm and the length of a diagonal is 7.5 cm.
Solution:
Steps of constructions are as follows:
Step 1: Draw a line segment AB of length 5 cm.
Step 2: Using protractor, draw perpendicular BX to AB through B.
Step 3: Using compass, draw an arc with centre A and radius 7.5 cm.
Let arc and BX intersect at point C.
Step 4: Using protractor, draw perpendicular AY to AB through A, and perpendicular CZ to BC through C.
Let AY and CZ intersect at point D. ABCD is the required rectangle.
Question 8.
The distance between points A and B is 7 cm. Now, mark the points that are:
(i) 3 cm away from both A and B
(ii) 4.5 cm away from both A and B
Solution:
(i) Step 1: Using a compass, draw a circle with centre A and radius 3 cm.
Step 2: Using a compass, draw a circle with centre B and radius 3 cm.
Since the two circles do not intersect, there is no point that is exattly 3 cm away from both point A and point B.
(ii) Step 1: Using a compass, mark arcs (with centre A and radius 4.5 cm) above and below AB.
Step 2: Using a compass, mark arcs (with centre B and radius 4.5 cm) above and below AB such that they cut the arcs constructed in step 1.
Let above arcs intersect at point P and below arcs intersect at point Q.
P and Q are at distance of 4.5 cm from points A and B.
Question 9.
Construct a square with the following side length:
(i) 3 cm
(ii) 4.5 cm
(iii) 5.5 cm
Solution:
(i) Steps of construction of a square with side 3 cm are as follows:
Step 1: Draw a line segment AB = 3 cm using a ruler.
Step 2: At points A and B, construct two perpendiculars using a protractor.
Step 3: From points A and B, mark points D and C on the perpendiculars respectively such that AD = BC = 3 cm.
Step 4: Join CD to complete the square ABCD.
(ii) Follow the same steps for square with side 4.5 m as in (i).
(iii) Follow the same steps for square with side 5.5 cm as in (i).
Question 10.
Construct a rectangle of the following side lengths:
(i) 3 cm and 5 cm
(ii) 4 cm and 6.5 cm
(iii) 5.5 cm and 7.5 cm
Solution:
(i) Steps of construction of a rectangle with side lengths 3 cm and 5 cm are as follows:
Step 1: Draw a line segment AB = 5 cm using a ruler.
Step 2: At points A and B, draw perpendiculars to AB using a protractor.
Step 3: Mark points D and C on both the perpendiculars respectively such that AD = BC = 3 cm.
Step 4: Join CD to complete the rectangle ABCD.
(ii) Follow the same steps for rectangle with side lengths 4 cm and 6.5 cm as in (i).
(iii) Follow the same steps for rectangle with side lengths 5.5 cm and 7.5 cm as in (i).
Question 11.
Construct a square within a rectangle such that the centre of the square is the same as the centre of the rectangle. The sides of the rectangle are 7 cm and 4 cm.
Solution:
As square lies inside the rectangle, the side of the square is equal to the smaller side of the rectangle.
So, side of the square = 4 cm
Steps of construction:
Step 1: Draw a rectangle ABCD such that AB = 7 cm and BC = 4 cm.
Step 2: With the help of ruler, mark P, Q, R and S as the mid-points of sides AB, BC, CD and DA.
Step 3: Join PR and QS. Their point of intersection 0 is the centre of the rectangle.
Step 4: With P as centre and OP as radius, mark arcs on both sides ofP intersecting AB at E and F respectively.
Step 5: Similarly, with R as centre and OR (= OP) as radius, mark arcs on both sides of R intersecting CD at G and H respectively. Then, join HE and GF.
Thus, EFGH is the required square whose centre is same as the centre of rectangle ABCD.
Question 12.
Recreate the following combinations of square and hole:
Centre of the hole is the same as the centre of the square containing it.
Solution:
Side length of square is not given. You can take any side length of your choice, say 12 cm.
Step 1: Construct a square ABCD of side 12 cm.
Step 2: Using a ruler, mark points P, Q, R and S as the mid-points of AB, BC, CD and DA respectively.
Join PR and QS. Let 0 be their point of intersection.
Step 3: Draw diagonals AO and PS. Let X be their point of intersection. Point X is the centre of square APOS.
Step 4: Using a compass, draw a circle with centre X and radius less than half of the side length of the square APOS.
Step 5: Draw circles in squares PBQO, OQCR and ORDS (same as step 4). Name their centres as Y, Z and W respectively.
Playing with Constructions Class 6 Long Question Answer
Question 1.
Recreate the following combinations of square and curves:
Note: All the 4 arcs bulge uniformly from each side.
Solution:
Side length of square is not given. We can take any side length of our choice, say 6 cm.
Steps of constructions are as follows:
Step 1: Construct a square ABCD of side 6 cm.
Step 2: Using a ruler, mark points P, Q, R and S as the mid-points of AB, BC, CD and DA respectively.
Step 3: Join PR and produce it in both directions. Also, join QS and produce it in both directions.
Step 4: Mark points X,Y, Z and W such that
XP = YQ = ZR = WS ≥ \(\frac{A B}{2}\)
Step 5: Using compass, draw an arc with X as centre and AX as radius such that it passes through A and B.
Similarly, draw arcs with centres Y, Z and W and radius YB, ZC and DW respectively.
The figure so obtained is the required square with curves.
NOTE: If we take XP = YQ = ZR = WS < \(\frac{A B}{2}\), then adjacent arcs will intersect at two points. But that is not the case.
Question 2.
A rectangular orchard of size 20 m × 15 m is to be planned using the square and diagonal planting systems. Each tree requires 5 m spacing. The planner uses only a compass, ruler and protractor to design accurate layouts based on geometric principles.
Based on the above information, answer the following questions:
i) How many trees can be planted using the square system, with 5 in spacing on a 20m × 13 in held?
(ii) Two trees are 6 m apart. How can you find a point that is 4.5 m away from both? Explain using compass method.
Solution:
(i) Given, field dimensions = 20 m × 15 m
Along the 20 m side (length), trees can be planted at the positions 0 m, 5 m, 10 m, 15 m, and 20 m. That is, there are a total of five positions along the 20 m side.
Therefore, the total number of columns of trees = 5.
Along the 15 m side (breadth), trees can be planted at the positions 0 m, 5 m, 10 m, and 15 m. That is, there are a total of four positions along the 15 m side.
Therefore, the total number of rows of trees = 4.
Hence, total trees = 5 × 4 = 20 trees
(ii) Given, two trees are 6 metres apart, we can find a point that is 4.5 metres away from both by following these steps:
Step 1: Mark two points A and B, 6 metres apart and join them using ruler.
Step 2: Set your compass to 4.5 cm.
Step 3: With A as the centre, draw two arcs, one above and one below the line segment AB.
Step 4: With B as centre, draw arcs of the same radius to intersect the previous arcs.
Step 5: Label the intersection points as P and 0.
Hence, points P and Q are both exactly 4.5 m away from A and B.
Question 3.
Recreate the following House:
Note: All the lines forming the border of the house are of length 4 cm.
Solution:
Steps of construction are as follows:
Step 1: Draw a line segment AB = 4 cm.
Step 2: Construct perpendiculars AD and BC to AB such that BC = AD = 4 cm.
Step 3: With D as centre draw an arc of length 4 cm. Also, draw an arc of same length with C as centre meeting previously drawn arc at point P.
Step 4: Join CP and PD, and with P as centre and CP as radius draw an arc joining C and D. The figure so formed represents a house whose all sides are of length 4 cm.
Step 5: Mark points E and F on AB such that
AE = BF = \(\left(\frac{4-1}{2}\right)\) cm = \(\frac{3}{2}\) cm = 1.5 cm.
Step 6: Draw perpendiculars EH and EC on AB at E and F respectively such that EH = FG = 2 cm. Join GH.