By using Ganita Prakash Book Class 6 Solutions and Chapter 8 Playing with Constructions Class 6 NCERT Solutions Question Answer, students can improve their problem-solving skills.
Class 6 Maths Chapter 8 Playing with Constructions Solutions
Playing with Constructions Class 6 Solutions Questions and Answers
8.1 Artwork Figure it Out (Page 191)
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:
The radius of the half-circle is 2 cm because the length of AB is 8 cm and AX is half of AB.
AX is the diameter of a circle. And the radius is half of the diameter
Question 2.
Take a central line of a different length and try to draw the wave on it.
Solution:
Do it yourself.
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:
Draw a horizontal line segment AB of any length, say 6 cm.
To draw the wave smaller than a half circle, first mark the mid-point ‘X’ of AB and then the mid points Y and Z of AX and XB respectively.
For drawing the first curve take the centre P, \(\frac{1}{2}\) cm below Y and radius = PA or PX.
Now, align the second wave right next to the first one but in opposite directions, by taking a point Q, \(\frac{1}{2}\) cm above Z and radius XQ or BQ.
8.2 Squares and Rectangles Figure it Out (Page 194)
Question 1.
Draw the rectangle and four squares configuration (shown in the figure 8.3 on page 192 of NCERT Textbook) 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 the rectangle on a dot paper with measurement 5×3 units, where space between two dots is considered as 1 unit.
Now, to place four squares symmetrical around the rectangle, first we draw one square, which is 1 unit upward and 1 unit leftward to rectangle. And one square, which is 1 unit upward and 1 unit rightward to the rectangle.
Similarly, draw two square below the rectangle as we drawn above.
Question 2.
Identify if there are any squares in this collection. Use measurements, if needed.
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?
Solution:
Fig. I: In this figure, AB and BC are not equal. So, ABCD cannot be a square.
Fig. II: In this figure, ∠BAD is not equal to 90°. So, ABCD cannot be a square.
Fig. III: In this figure, counting dots between sides, we find that AB, BC, CD, and DA are all equal sides. Also, the position of the dots on the sides shows, that each angle of ABCD is 90°.
∴ ABCD is a square.
Fig. IV: In this figure, counting dots between sides, we find that AB, BC, CD, and DA are all equal sides. Also, using a protractor, we find that each angle of ABCD is 90°.
∴ ABCD is 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:
In all the squares, the sides are equal in length and all vertex angles are 90°.
In all the rectangles, the opposite sides are equal in length and each vertex angle is 90°.
8.3 Constructing Squares and Rectangles 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:
Draw a line segment PQ of length 6 cm.
Mark a point to draw a perpendicular to PQ through P.
Mark point S on the perpendicular such that PS = 4 cm using a ruler.
Similarly, draw a perpendicular to PQ through Q and mark point R on the perpendicular such that QR = 4 cm. Joint RS.
Thus, PQ = SR = 6 cm and PS = QR = 4 cm and ∠P = ∠Q = ∠R = ∠S = 90°.
∴ PQRS is a rectangle, as it satisfies all the properties of rectangle.
Question 2.
Draw a rectangle of sides 2 cm and 10 cm. After drawing check if it satisfies both the rectangle properties.
Solution:
We shall draw a rectangle of the form shown in Fig. 1.
Step 1. Using a ruler, draw a line AB equal to 10 cm. (Fig. 2).
Step 2. Using a protractor, draw perpendicular lines at A and B (Fig. 3).
Step 3. Using a ruler, mark point P on the perpendicular at A such that AP = 2 cm. Using a ruler, mark point Q on the perpendicular at B such that BQ = 2 cm. (Fig. 4)
Step 4. Join P and Q using a ruler. Erase the lines above P and Q. (Fig. 5)
Step 5. Using a ruler, verify that PQ is of length 10 cm.
Using a protractor, verify that ∠P and ∠Q are 90° each.
Step 6. We have:
(i) AB = PQ = 10 cm and AP = BQ = 2 cm
(ii) ∠A = ∠B = ∠P = ∠Q = 90°.
Step 7. ABQP in Fig. 5 is the required rectangle of sides 2 cm and 10 cm.
Question 3.
Is it possible to construct a 4-sided figure in which-
(i) All the angles are equal to 90° but
(ii) Opposite sides are not equal?
Solution:
Step 1. Using a ruler, draw a line AB equal to 6 cm, say. (Fig. 1).
Step 2. Using a protractor, draw perpendicular lines at A and B (Fig. 2).
Step 3. Using a ruler, mark point P on the perpendicular at A such that AP = 4 cm.
Using a ruler, mark point Q on the perpendicular at B such that BQ = 2 cm, which is not equal to AP. (Fig. 3).
Step 4. In Fig. 3, the opposite sides of AP and BQ are not equal. Join P and Q using a ruler. Erase the lines above P and Q (Fig. 4).
Step 5. Using a protractor, we find that neither ∠P nor ∠Q is 90°.
Step 6. We conclude that it is not possible to construct a 4-sided figure in which all angles are 90° and opposite sides are not equal.
8.4 An Exploration in Rectangles 8.5 Exploring Diagonals of Rectangles and Squares Construct (Page 211)
Question 1.
Construct a rectangle in which one of the diagonals <1 divides the opposite angles into 50° and 40°.
Solution:
Draw a horizontal line segment AB. This will be one side of the rectangle.
(i) At point A, use a protractor to measure and draw AX a 50° angle.
(ii) At point B, measure and draw a 90° angle. Draw a line segment BC extending from B at this angle meeting AX at C.
(iii) At Point A and C, draw a 90° angle which meets at the point D.
ABCD is the required rectangle. Here, diagonal AC divides the opposite angles A and C into 50° and 40°.
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:
Do it yourself.
Sides of the rectangle would be equal in this case.
Question 3.
Construct a rectangle one of whose sides is 4 cm and the diagonal is of length 8 cm.
Solution:
We shall draw a rectangle of the form shown in Fig. 1.
Step 1. Using a ruler, draw a line AB equal to 4 cm. (Fig. 2)
Step 2. Using a protractor, draw a perpendicular line to AB at A and B (Fig. 3)
Step 3. With the centre at A and a radius equal to 8 cm, draw an arc to intersect the perpendicular at B. Similarly, draw an arc of radius 8 cm with a centre at B to intersect perpendicular at A. (Fig. 4)
Step 4. Join the points of intersection of arcs by PQ. (Fig. 5)
Step 5. Erase the extra lines in Fig. 5 (Fig. 6)
Step 6. Fig. 6 is the required rectangle with one side equal to 4 cm and diagonals of length 8 cm.
Question 4.
Construct a rectangle one of whose sides is 3 cm and the diagonal is of length 7 cm.
Solution:
Do it by yourself.
8.6 Points Equidistant from Two Given Points Construct (Page 215)
Question 1.
Construct a bigger house in which all the sides are of length 7 cm.
Solution:
Steps
(i) Draw a horizontal line DE of length 7 cm.
(ii) At point D, draw a perpendicular line upwards using your scale or set square. Measure 7 cm on this line and mark point B.
(iii) Similarly, at point E, draw a perpendicular line upwards and ma$k point C at a distance of 7 cm.
(iv) Using your compass, set a radius of length 7 cm. With B as the center, draw an arc. With the same radius, and C as the center, draw another arc to intersect the first arc. Mark this intersection as point A.
(v) Draw line segments AB and AC to complete the triangle.
(vi) With A as a centre and AB as radius draw an arc BC.
(vii) On the base DE, draw a rectangle of sides 2 cm and 1 cm.
Hence, ABDEC is the required house.
Question 2.
Try to recreate ‘A Person’, ‘Wavy Wave’, and ‘Eyes’ from the section Artwork, using ideas involved in the ‘House’ construction.
Solution:
Do it by yourself.
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, there is a four-sided figure where all sides are equal in length but it is not a square.
This figure is called a rhombus.
First draw a line segment of any length say AC.
Taking the radius more than half of the length of the line segment AC and end points A and C as centres, draw arcs above and below AC. Let they intersect at points B and D respectively.
These points are the other two vertices of the rhombus.
Join AD, AB, BC and CD.
∴ ABCD is a rhombus.