Free access of the complete Ganita Prakash Book Class 6 Solutions and Chapter 8 Playing with Constructions Class 6 NCERT Solutions Question Answer are crafted in simple format to align with the latest CBSE syllabus.

## Class 6 Maths Chapter 8 Playing with Constructions Solutions

### Playing with Constructions Class 6 Solutions Questions and Answers

**8.1 Artwork Construct (Page No. 190 – 191)**

Question 1.

What radius should be taken in the compass to get this half circle? What should be the length of AX?

Solution:

The diameter AX of the half circle shown in the figure is half length of line segment AB. So, AX = \(\frac{8}{2}\) = 4 cm.

To draw this half circle, we should take radius half of AX, i.e., 2 cm in the compass.

Question 2.

Take a central line of a different length and try to draw the wave on it.

Solution:

Let us take AB to be the central line such that the length of AB is 12 cm. That is, AB = 12 cm.

Now, the first wave is drawn as a half circle, using the diameter half of the central line AB, i.e., the radius half 6 of AX = \(\frac{6}{2}\) cm = 3 cm 2

Now, the second wave is drawn as the half circle with radius half of XB = 6 cm, i.e., of 3 cm in opposite direction to the first wave.

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 comers in the dot grid?

Solution:

In figure A, all sides are 4 units in the dot grid and all angles are 90°. So, figure A represents a square in the dot grid.

Question 3.

Draw at least 3 rotated squares and rectangles on a dot grid. Draw them such that their comers are on the dots. Verify if the squares and rectangles that you have drawn satisfy their respective properties.

Solution:

We can verify the properties of square and rectangles on the dot grid.

Check the length of the sides which align with the grid lines, by counting the number of dots between the comers or measure using a ruler. Further find the measure of each angle using a protractor and check whether it is 90°, i.e., the right angle or not.

**8.3 Constructing Squares and Rectangles Construct (Page No. 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:

Do it yourself.

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:

Let us draw a line segment PQ of any length.

Now, mark a point to draw a perpendicular to PQ through P. and mark point S on the perpendicular at any length using ruler.

Again repeat the previous step at points Q and S, to get the perpendiculars on PQ and PS respectively. Let the perpendiculars drawn intersect at point R.

Now, measure ∠R, which is 90°.

So, ∠P = ∠Q = ∠R = ∠S = 90°

Also, measure PQ, QR, RS and PS.

Here, PQ = SR and PS = QR

∴ It is not possible to construct a 4-sided figure with the given conditions.

**8.4 An Exploration in Rectangles Construct (Page No. 201 – 203)**

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 comers of the square and the outer rectangle?

Solution:

Draw a rectangle PQRS of sides 8 cm and 4 cm.

Next, draw lines LN and MO from the midpoints of opposite sides such that they intersect side PQ at L, QR at M, RS at N, and PS at O respectively.

Since the centre of square and rectangle is same and the width of rectangle is 4 cm, so we draw a square of side 4 cm with the centre of rectangle and named as ABCD.

∴ The side length of the square is 4 cm and the distance between the comers of the square and the outer rectangle = PL – AL = 4 cm – 2 cm = 2 cm.

Question 2.

Falling Squares

Now, try this.

Solution:

Draw a square of side 4 cm.

Again, draw a square of side 4 cm at one comer of the previous square.

Repeat the previous step, we get

Draw the second figure by yourself.

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:

Draw a square of sides 4 cm each and cut it into 4 smaller squares with sides 2 cm each.

Again cut the one smaller square with side 2 cm each into 4 smaller squares with side 1 cm each.

Repeat the previous step in 2 more smaller squares of side 2 cm each and leave one smaller square as it is.

Now, out of 12 smaller squares, draw a diagonal in 11 smaller squares and shade its one part. Then, we get the required figure.

Question 4.

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.

Solution:

Draw a square with side 4 cm.

Determine the centre by drawing the diagonals of the square.

Now, from the centre of square, draw a circle with radius 1 cm.

We get the required figure.

Question 5.

Square with more Holes

Solution:

Do it Yourself.

Question 6.

Square with curves

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:

First draw a square with 8 cm sidelengths named ABCD.

Now, mark the midpoints of each side of the square as L, M, N and O on sides AB, BC, CD and DA respectively. Also, join diagonals AC and BD. Let they intersect at P.

Further, draw lines passing through L and N and extend it to the points W and Y such that PL = LW = PN = NY. Similarly, extend line OM to the points X and Z.

Using these points as centres draw four circular axes with a radius equal to WA. Each arc should be drawn inside the square, connecting two adjacent vertices.

Thus, we get the required figure.

**8.5 Exploring Diagonals of Rectangles and Squares Construct (Page No. 211)**

Question 1.

Construct a rectangle in which one of the diagonals divides the opposite angles into 50° and 40°.

Solution:

Step 1: Draw a line with an arbitrary length.

Step 2: Draw a perpendicular on AB through B and mark D on the perpendicular at any length using ruler.

Step 3: Put protractor at point A and mark point C on line BD such that ∠BAC = 50°.

Step 4: Draw a line perpendicular to AB at A and mark the point E such that AE = BC.

Thus, ABCE is the required rectangle with ∠CAB = 50° and ∠EAC = 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:

First draw the base AB of length 4 cm.

Now, draw a perpendicular to line AB at point B. Let us call this line m.

Now, using a ruler, take the distance of 8 cm in a compass, and from A, mark an arc cutting line m at C. Join AC.

Now, construct perpendiculars to AB and BC at points A and C respectively. The point where these perpendiculars intersect is the fourth point D.

Thus, ABCD is a rectangle with side AB = 4 cm and diagonal AC = 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 yourself.

**8.6 Points Equidistant from Two Given Points Construct (Page No. 215)**

Question 1.

Construct a bigger house in which all the sides are of length 7 cm.

Solution:

First draw a square of side 7 cm, named as ABCD.

Now, using a compass draw arcs above the side AB of radius 7 cm from points A and B. Let the arcs intersect at point E.

Joint A to E and B to E by straight lines

Now, take 7 cm radius in the compass and from E, draw the arc touching A and B as shown in the figure.

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 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.